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+{"text":"\\section{Hausdorff dimension in the collapsed and branched polymer phase}\n\\begin{figure*}[!htb]\n\\centering\n    \\begin{subfigure}[b]{0.45\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/BPexample.pdf}   \n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.45\\textwidth}  \n    \\centering \n    \\includegraphics[width=\\textwidth]{figures\/CPexample.pdf}  \n    \\end{subfigure}\n    \\caption\n    {\\small Data collapse of the three volume $N_3(\\rho)$ with scaled distance is consistent with (a) $D_H=2.0$ in the BP phase, (b) $D_H=12.0$ in the collapsed phase.} \n        \\label{fig_BP_CP_DH}\n\\end{figure*}\n\\end{comment}\n\n\n\n\\section{Relative lattice spacing \\label{appendix1}}\n\\begin{figure}[!ht]\n\t\\vspace{12pt}\n   \\centering\n   \\begin{subfigure}{0.45\\textwidth}\n       \\centering\n       \\includegraphics[width=\\textwidth]{DS\/lnp_witherror_near_critical.pdf}\n       \\subcaption{\\label{fig_ret_prob}}\n   \\end{subfigure}\n   \\vfill\n   \\begin{subfigure}{0.45\\textwidth}\n       \\centering\n       \\includegraphics[width=\\textwidth]{DS\/lnp_sigma_near_critical.pdf}\n       \\subcaption{\\label{fig_rel_a}}\n   \\end{subfigure}\n    \\caption{ Return probability at different points in the critical line at lattice volume $V=32$ k (a) with respect to diffusion step ($\\sigma$), (b) with respect to rescaled diffusion step ($\\sigma_r$). The scaling allows to find relative lattice spacing along the transition line. Associated error-bars are not shown in the right-hand figure to demonstrate the superimposed data from different points in the transition line.  \\label{relative} }\n\\end{figure}\n\n\\begin{table}[!ht]\n\t\\centering\n\t\\begin{tabular}{|c| c|c| c|c|c|c|  c|c|c|c|}\\hline\n\t\t$\\kappa_c$ & -0.90& -0.7375 & 1.5625 & 2.0 &2.5 & 3.0 &3.5  & 4.0 &4.5 &5.0 \\\\ \\hline\n\t\t$a_r$ & 1.5 & 1.475 & 1.135 & 1.05 &1 &0.935&0.92&0.895&0.87 & 0.86\\\\ \\hline \n\t\\end{tabular}\n\t\\caption{Relative lattice spacing along the transition line as $\\kappa$ is varied.}\n\t\\label{tab_a_r}\n\\end{table}\nIn this work, we did not attempt to perform a precise measurement of the renormalized gravitational constant which determines the absolute lattice spacing. Two different methods of finding the gravitational constant in the context of the Euclidean dynamical triangulation can be found in the two recent papers by Laiho \\textit{et.el.} \\cite{dai2021newtonian,bassler2021sitter}. We have\nused the relative lattice spacing as obtained from the return probability in our work.\nIn fig.~\\ref{fig_ret_prob} we show the return probability at several different\npoints along the critical curve for $V=32$ K and in fig.~\\ref{fig_rel_a} and show how\nthese curves can be collapsed onto a single curve by rescaling the step size\n$\\sigma$. Rescaling of the step $\\sigma=\\sigma_r a_r^2$ can be interpreted as yielding a relative lattice spacing $a_r$ as $\\kappa$ varies along the critical curve. Values of the relative lattice constant $a_r$ are noted in the Table~\\ref{tab_a_r} and are consistent with the previous work by Laiho \\textit{et. el.}. Namely they reveal that as $\\kappa$ approaches infinity the\ncorresponding lattices get finer \\cite{laihoLatticeQuantumGravity2017}. Hence, for a fixed target volume $V$, the physical volume is smaller at larger $\\kappa_c$ and it is likely that the results obtained would suffer greater finite size effect in that region.\n\n\n\n\\section{Introduction}\nThere are many proposals for quantizing four dimensional gravity - see\nthe reviews \\cite{ashtekar2021short,loll2019quantum,weinberg1979ultraviolet,niedermaier2006asymptotic}. In this paper we explore one such approach known as Euclidean Dynamical Triangulation (EDT) in which the continuum path integral is replaced by a discrete sum over\nsimplicial manifolds. This approach is similar in spirit\nto the Causal Dynamical Triangulation (CDT) program \\cite{loll2019quantum,ambjorn2013causal} after relaxing the\nconstraint that each triangulation admit a discrete time\nslicing. In practice we restrict to triangulations\nwith equal edge lengths and fixed topology. In addition we only\ninclude so-called combinatorial triangulations in the discrete path integral\nwhich guarantees that the neighborhood\nof each vertex is homemorphic to a $4-$ball. This ensures\nthat any p-simplex in the triangulation is uniquely specified in\nterms of its vertices. This differs from\nrecent work by Laiho \\textit{et al.} which utilizes an ensemble of degenerate triangulations\nand a different measure term \\cite{laihoEvidenceAsymptoticSafety2011,laihoLatticeQuantumGravity2017,daiNewtonianBindingLattice2021,basslerSitterInstantonEuclidean}. Our work is also complementary to that \nof Ambjorn \\textit{et al.} \\cite{ambjornEuclidian4dQuantum2013} who employ the same class of triangulations but a different\nmeasure term. \n\nThe goal of our work has been to provide a detailed picture of the phase diagram of the model\nand the location of possible phase transitions by simulating the model over a fine grid in\nthe two dimensional parameter space for three lattice volumes ranging \nup to $N_4=32,000$ 4-simplices. We find evidence for a single critical line\nseparating a crumpled from a branched polymer phase consistent with all\nearlier studies of similar models. \nIn addition to certain bulk observables we \nhave focused\nour attention on the Hausdorff and spectral dimensions along this critical line and are able\nto compute these both along and transverse to this critical line in some detail.\n\n\\section{The lattice model}\n\n\\begin{figure*}[!htb]\n\t\\centering\n\t\\hspace{20pt}\n\\subfloat[\\label{fig_chiN0}]{%\n\t\\centering\n  \\includegraphics[width=.45\\textwidth]{figures\/chi.N0.b0.25.jk.pdf}%\n}\\hfill\n\\subfloat[\\label{fig_chilogq}]{%\n\t\\centering\n  \\includegraphics[width=.45\\textwidth]{figures\/chi.logq.b0.25.jk.pdf}%\n}\n\n\\hspace{20pt}\n\\caption{Susceptibility plots (a) $\\chi_{N_0}$ and (b) $\\chi_{\\log{q}}$ for $V=32K$ are shown. The peaked structure near $\\kappa \\sim 1.6$ indicates a phase transition.}\n\\label{fig:sus}\n\\end{figure*}\n\\begin{figure*}[!htb]\n        \\centering\n        \\begin{subfigure}[b]{0.45\\textwidth}\n            \\centering\n            \\includegraphics[width=\\textwidth]{figures\/MCtime.N0.k1.60.b0.25.png}   \n        \\end{subfigure}\n        \\hfill\n        \\begin{subfigure}[b]{0.45\\textwidth}  \n            \\centering \n            \\includegraphics[width=\\textwidth]{figures\/freq.N0perVol.k1.60.b0.25.pdf}  \n        \\end{subfigure}\n        \\caption\n        {\\small First order nature of the transition at $\\beta=0.25$ can be see from the (a) Monte Carlo time series for $N_0\/N_4$ and (b) double peak structure of the probability density of $N_0\/N_4$.} \n        \\label{fig:mc}\n    \\end{figure*}\nThe partition function of the model for the pure gravity takes the form\n\\begin{equation}\n    Z=\\sum_T \\rho(T) e^{-S}\\label{partition}\n\\end{equation}\nwhere the discrete action $S$ is given by\n\\begin{equation}\n    S=-\\kappa N_0+\\lambda N_4+\\gamma(N_4-V)^2,\n\\end{equation}\nwhere the sum runs over all abstract triangulations\nwith fixed (here spherical) topology~\\footnote{Numerical evidence has been presented in previous studies that the number of possible 4d triangulations of fixed spherical topology is exponentially bounded \\cite{catterall1996baby,ambjorn1994exponential} and\nhence can be controlled by a bare cosmological constant term.}\nThe first two terms in the action depending on the number of vertices $N_0$\nand the number of four simplices $N_4$ arise from using Regge calculus \\cite{regge1961general,thorne2000gravitation} to\ndiscretize the continuum Einstein-Hilbert action with $\\kappa$ playing the role\nof the bare Newton constant and $\\lambda$ a bare cosmological constant. \nThe third term plays an auxiliary role in effectively fixing the target volume to $V$ by\ntuning $\\lambda$ while still allowing for \nsmall fluctuations $\\delta V\\sim \\frac{1}{\\sqrt{\\gamma}}$.\n\nThe central assumption in this approach to quantum gravity\nis that the sum over triangulations\nreproduces, in some appropriate continuum limit, the ill-defined continuum path\nintegral over metrics modulo diffeomorphisms. In two dimensions this prescription is known to reproduce known results\nfor 2d gravity from Liouville theory and matrix models \\cite{boulatov1986analytical,kazakov1986ising} but in higher\ndimensions it is merely a plausible ansatz. \n\nThe assumption\nof most recent works\nis that an additional measure term $\\rho(T)$, which\ndepends on local properties of the triangulation, is needed to ensure this correspondence with continuum gravity remains true \\cite{ambjornEuclidian4dQuantum2013,laihoLatticeQuantumGravity2017}.  Here we employ a new form\n\\begin{equation}\n    \\rho(T)=\\prod_{i=1}^{N_0}q_i^\\beta,\n\\end{equation}\nwhere $q_i$ denotes the number of simplices sharing vertex $i$ and $\\beta$ is\na new parameter. This is similar to the local measure term used in previous studies \\cite{laihoLatticeQuantumGravity2017,ambjornEuclidian4dQuantum2013}. It is conjectured\nthat tuning the coupling to such an operator is necessary to restore continuum\nsymmetries and approach a fixed point where a continuum limit describing quantum\ngravity can be taken \\cite{ambjornEuclidian4dQuantum2013}. \n\nOur work is focused on examining the phase structure of the model\nin the $(\\kappa,\\beta)$ with the goal of searching for critical behavior and locating\na region where such a continuum limit can be taken.\n\n\\section{Phase Structure}\nWe employ a Monte Carlo algorithm to sample the sum over random\ntriangulations \\cite{catterall1995simulations}. Five elementary \nlocal moves (``Pachner moves\") which, iterated appropriately\nare known to be sufficient to\nreach any part of the triangulation space. \n\nTwo of the simplest observables that can be used to locate the transition are\nthe node and measure susceptibilities which are defined by\n\\begin{align}\n    \\chi_{N_0}&=\\frac{1}{V}\\left(\\left\\langle N_0^2\\right\\rangle-\\left\\langle N_0\\right\\rangle^2\\right)\\\\\n    \\chi_{\\ln{q}}&=\\frac{1}{V}\\left(\\left\\langle Q^2\\right\\rangle-\\left\\langle\n    Q\\right\\rangle^2\\right)\n\\end{align}\nwith $Q=\\frac{1}{N_0}\\sum_{i=1}^{N_0}\\ln{q_i}$.\nIn fig~\\ref{fig:sus} we show these as a function of\n$\\kappa$ at $\\beta=0.25$ for a lattice of (average) volume $N_4=32,000$. The peak in both\nquantities indicates the presence of a phase transition.\n\nIn fig.~\\ref{fig:mc} we show the Monte Carlo time evolution\nof the vertex number $N_0$ and its associated probability distribution for a $V=32,000$ simplex simulation close to\nthe critical line at $\\beta=0.25$. The tunneling behavior\nin the Monte Carlo time series together with the double peak structure in \nthe probability distribution for the number of vertices $P(N_0)$ constitute strong\nevidence that the transition is first order in this region. This precludes\na continuum limit and indeed the observation of a similar structure\nat $\\beta=0$ was the original motivation\nfor introducing a measure term.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{figures\/chi.logq.k5.0.b.pdf}\n    \\hspace{10pt}\n    \\caption{At large $\\kappa$ two distinctive peaks are observed in the susceptibility plots. Position of the critical points are shown with the vertical lines.}\n    \\label{broadpeak}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering \n\t\\includegraphics[width=0.45\\textwidth]{figures\/1ravg_comparison.pdf}   \n\t\\caption{$\\beta$ dependence of the average radius $r_{\\rm avg}$ at different fixed values of $\\kappa$, at a target volume $V=32$K. Vertical lines denote the position of the critical points $\\hat{\\beta}_c$ at different Gravitational constants $\\kappa$.}\n\t\\label{fig_ravg}\n\\end{figure}\n\n\\begin{table}[!htb]\n\t\\centering\n\t\n\t\\renewcommand{\\arraystretch}{1.5}{\n\t\t\\begin{tabular}{\n\t\t\t\t|>{\\centering\\arraybackslash}p{2.5cm}||\n\t\t\t\t>{\\centering\\arraybackslash}p{2.5cm}|\n\t\t\t\t>{\\centering\\arraybackslash}p{2.5cm}|\n\t\t\t} \n\t\t\t\\hline\n\t\t\t$\\bm{\\beta}$ & $\\bm{\\kappa_c}$  &  $\\bm{\\hat{\\kappa}_c} $\\\\ \\hline\n\t\t\t1.00 & -0.89(1)  & -0.894(6) \\\\ \\hline\n\t\t\t0.50 & 0.75(1) &  0.756(6)\\\\ \\hline \n\t\t\t0.25 & 1.61(2)  & 1.606(6) \\\\ \\hline \\hline\n\t\t\t$\\bm \\kappa$ & $\\bm \\beta_c$ & $\\bm{\\hat{\\beta}_c}$\\\\ \\hline\n\t\t\t2.0 & 0.14(1) & 0.144(6)    \\\\ \\hline\n\t\t\t2.5 & 0.00(1) & 0.006(6) \\\\ \\hline\n\t\t\t3.0 & -0.13(1) & -0.13(1)     \\\\ \\hline\n\t\t\t3.5 & -0.25(1) & -0.244(6) \\\\ \\hline\n\t\t\t4.0 & -0.36(1) & -0.35(1)    \\\\ \\hline\n\t\t\t4.5 & -0.46(1)  & -0.46(1)  \\\\ \\hline\n\t\t\t5.0 & -0.56(1)  & -0.56(1)\\\\ \\hline\n\t\t\\end{tabular}\n\t}\n\t\\caption{Pseudo-transition point $\\kappa_{c}$ ($\\beta_c$) obtained from fixed $\\beta$ ($\\kappa$) scan of the susceptibilities at target volume $V=32$ k vs corresponding\n\t\testimates of the critical point $\\hat{\\kappa}_c$ ($\\hat{\\beta}_c$) determined\n\t\tfrom the average radius $r_{\\rm avg}$. } \\label{tab_TP} \n\\end{table}\n\nAs we increase $\\kappa$ we observe that \nthe latent heat of the transition, as measured by the separation in\nthe two peaks in the probability distribution $P(N_0)$ decreases and the structure\nof the susceptibility plots changes. If one fixes $\\kappa$\none observes a broad peak centered at $\\beta_{c1}$ followed by a much narrower peak at $\\beta_{c2}$ with $\\beta_{c2}> \\beta_{c1}$, Fig.~\\ref{broadpeak}. \nFor $\\beta>\\beta_{c2}$ the system is clearly in the branched polymer phase while for $\\beta<\\beta_{c1}$ the system is clearly in the crumpled phase. The separation $\\Delta \\beta$ between the two critical points narrows down as the volume is increased. In our work we have used $\\beta_{c2}$ as our best estimate for the true critical point $\\beta_c$.\n\nTo complement this determination of the critical point we have also studied the mean radius of the discrete geometry.\nThis is defined by\n\\begin{align}\n    r_{\\mathrm{avg}}=\\frac{1}{N_4}  \\left\\langle  \\sum_r r \\,N_{3}(r)   \\right\\rangle_{T},\n\\end{align}\nwhere $N_3(r)$ is the\nnumber of four-simplices at geodesic distance $r$ \nmeasured on the dual lattice from some randomly chosen origin.\nIn fig.~\\ref{fig_ravg} we show a plot of the mean radius $r_{\\rm avg}$  vs $\\beta$ for\nseveral values of $\\kappa$.\nThe phase transition visible in the susceptibilities\nis clearly also seen in $r_{\\rm avg}$.  To find the critical coupling, we computed a numerical derivative of the radius as a function of\n$\\beta$ and identified the critical point $\\hat{\\beta}_{c}$ as the point where this derivative is maximal. A list of transition points derived from this observable are listed in the second column of the table \\ref{tab_TP} and shown to agree very well with the value $\\beta_{c}$ determined from the $\\chi(N_0)$ and $\\chi_{\\log{q}}$ susceptibilities. Notice for small $\\kappa$ we have fixed $\\beta$ and\nscanned the transition in $\\kappa$ while for large $\\kappa$ we have fixed $\\kappa$ and done a scan in $\\beta$ values\\footnote{This was motivated by the schematic phase diagram known from the earlier studies \\cite{laihoLatticeQuantumGravity2017}, where the transition line shows trend to asymptote to a negative $\\beta_c$ value at large $\\kappa_c$. It's true that there is no guarantee that the same trend will be followed in our analysis with the new measure term.}.\n\n\\section{Hausdorff dimension}\n\\begin{figure}[!htb]\n\\centering\n    \\begin{subfigure}[b]{0.45\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/BPexample.pdf}   \n    \\end{subfigure}\n    \\caption\n    {\\small Data collapse of the three volume $N_3(\\rho)$ with scaled distance is consistent with $D_H=2.0$ in the BP phase.} \n        \\label{fig_BP}\n\\end{figure}\nTo compute the Hausdorff dimension $D_H$ we assume that $N_3(r)$ takes the scaling form\n\\begin{equation}\n    N_3 (r)=N_{4}^{1 \/ D_{H}-1} f\\left(r\/N_{4}^{1 \/ D_{H}}\\right).\n\\end{equation}\nFitting to this form shows that\nthe Hausdorff dimension in the branched polymer phase is consistent with the value of $D_H=2$ (Fig.~\\ref{fig_BP})\nwhile in the collapsed phase, the extracted value of $D_H$ from such fits is large which is consistent with the continuum expectation of infinite $D_H$ \\cite{ambjorn1995scaling,coumbe2015exploring}.\nAt small distances, $N_3$ should grow as $\\sim r^{D_H-1}$ \\cite{ambjorn1995scaling}. In practice we have used this fact rather than data collapse on the scaling form\nto extract $D_H$ close to the critical line on our largest lattice by fitting \n\\begin{equation}\n    N_3=A\\,r^{D_H-1} +B.\n\\end{equation}\nFig~\\ref{N3fit} shows such a fit. The results presented are an ensemble average computed from 2000 thermalized configurations. The fit is performed at several fixed $\\beta$ and fixed $\\kappa$ to observe the variation in the Hausdorff dimension as we moved from the collapsed phase to the branched polymer phase. The value of the Hausdorff dimension is strongly influenced by\nthe distance from the critical line as can be seen in Fig.~\\ref{DHscan} which shows $D_H(\\beta)$ at a fixed $\\kappa=4.0$. From the rise of the value of DH towards the left, it is evident that as we probe deep into the collapsed phase, we get larger Hausdorff dimensions. Also clearly visible is the\nfact that deep in the BP phase on the right of the diagram the value approaches the known value of $D_H=2$.\n\n\n\\begin{figure}[!htbp]\n  \\includegraphics[width=.35\\textheight]{DH_DS\/SD_DH_fitsample.png}%\n \n\\caption{Fit of three volume data at small distance.\\label{N3fit}}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{DH_DS\/DHscan.k.4.0.pdf}\n    \\caption{Variation in Hausdorff dimension  $D_H$ with $\\beta$\n    at $\\kappa=4.0$ at $V=32$ K. Position of the critical coupling $\\beta_c$ derived from susceptibility is noted with the vertical line. }\n    \\label{DHscan}\n\\end{figure}\n\n\\begin{figure}[!htb]\n    \\centering\n    \\includegraphics[width=.45\\textwidth]{DH_DS\/DHvskc.pdf}%\n    \\caption{Fit of extracted Hausdorff dimension $D_H$ as a function of critical coupling along transition line.}\n   \\label{DHextrap}\n\\end{figure}\nThe value of  $D_H$ along the critical line is shown in fig.~\\ref{DHextrap} which also\nincludes a fit of the form\n\\begin{equation}\n    D_H=M\/(\\kappa+B)+D_{H,\\infty}.\n\\end{equation}\nHere, $M$ and $B$ are fit parameters and $D_H \\to D_{H,\\infty}$ as $\\kappa\\to\\infty$. We find $D_{H,\\infty}=4.06 \\pm 0.64$ which \nis consistent with the emergence of four dimensional de Sitter space in this limit.\n\nEncouraged by\nthis we have compared our three-volume distribution  near the\ncritical point at large $\\kappa$ with the (Euclidean) de-Sitter solution  \\footnote{A homogenous and isotropic universe as described by the FLRW metric}. The\nassociated three volume profile for the Wick rotated case takes the form of Eqn.~\\ref{desitter_3vol} \\cite{ambjorn2008nonperturbative,glaser2017cdt,laihoLatticeQuantumGravity2017}.  This is shown in Fig.~\\ref{deSitterfit} and indicates that the average geometry at small to intermediate\ndistances is indeed consistent\nwith de Sitter as $\\kappa$ gets large.\n\\begin{equation}\nN_{3} (r)=\\frac{3}{4} N_{4}^{3\/4} \\Gamma \\cos ^{3}\\left(\\frac{r-b}{s_{0} N_{4}^{1 \/ 4}} \\right) \\label{desitter_3vol}\n\\end{equation}\nHere, $s_0$, $\\Gamma$ and $b$ are fit parameters. One can think of\n$s_0$ as determining a relative lattice spacing for different values of\nthe $(\\kappa,\\beta)$. We find good matching of our data to the de-Sitter solution starting from a small distance $r$ up to about five steps beyond the maxima. The long tail of the distribution is likely\na finite size effect \\cite{laihoLatticeQuantumGravity2017}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{DH_DS\/1_DH_cosine_fit.k5.0.pdf}\n    \\caption{Fit to the de-Sittter solution of the three volume distribution data.}\n    \\label{deSitterfit}\n\\end{figure}\n\n\\section{Spectral Dimension}\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{DS\/fit2_b.-0.1375.k3.0.pdf}\n\t\\caption{Sample fit of the spectral dimension near the transition line in the phase space.}\n\t\\label{spectral_fit}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{DS\/32k.across.k4.0.Ds_inf_Ds_zero.pdf}   \n\t\\caption\n\t{\\small UV ($D_S(0)$) and IR ($D_S(\\infty)$) spectral dimension across transition at a fixed $\\kappa=4.0$. Vertical line denotes the position of the transition point and the two horizontal line denotes the $D_s$ value of 1.5 and 4 for comparison with the data.} \n\t\\label{fig_DSinf0_across}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\t\\centering \n\t\\includegraphics[width=0.45\\textwidth]{DS\/32k.along.Ds_inf_Ds_zero.pdf}  \n\t\\caption\n\t{\\small UV ($D_S(0)$) and IR ($D_S(\\infty)$) spectral dimension along the the transition line. Horizontal line denotes the $D_s$ value of 1.5 for the comparison.} \n\t\\label{fig_DSinf0_along}\n\\end{figure}\nAnother measure of\ndimension for a fluctuating geometry is called the ``spectral dimension\" $D_S$. \nIt can be computed\nfor a simplicial manifold $\\mathcal{M}$ using a random walk process. \nStarting from a randomly selected\nsimplex the random walk corresponds to successively moving from one simplex\nto one of its neighbors\nvia a randomly selected face. This process is then\niterated a large number of times. To compute the spectral dimension one records the\nnumber of times the walk returns to the starting simplex as a function of the diffusion\ntime (number of steps of the random walk).\nBy running several of these walks and averaging over starting\npoints and over the ensemble of configurations obtained at\nsome fixed $\\beta$ and $\\kappa$ \nwe can obtain the probability of returning to\nthe starting simplex $P_r(\\sigma)$ after $\\sigma$ steps. The spectral\ndimension is then defined from the relation:\n\\begin{equation}\nD_{S}(\\sigma)=-2 \\frac{d \\log \\left\\langle P_{r}(\\sigma)\\right\\rangle}{d \\log \\sigma}\n\\end{equation}\nThe return probability itself is a useful quantity which can be used to find the\nrelative lattice spacing at different points on the transition line \\cite{laihoLatticeQuantumGravity2017}. This is discussed in\nmore detail in appendix~\\ref{appendix1}.\n\nIn the branched polymer phase we observe $D_S=4\/3$ which is consistent\nwith theoretical expectations \\cite{jonsson1998spectral} while in the crumpled phase it is\nlarge. At the critical point we find $D_S$ is not well fitted by\na constant but instead runs with scale $\\sigma$.\nIn fig.~\\ref{spectral_fit} we show a plot of this running spectral dimension for $V=32K$ and $\\beta=-0.1375,\\,\\, \\kappa=3.0$. \\\\\n\nWe used 2000 thermalized configurations for the computation of the spectral dimension. Each random walk is performed up to 15000 steps and we choose 32000 randomly chosen sources per configuration. The fit is attempted over different ranges. Due to the finite volume of the lattices, the spectral dimension will increase and reach a maximum before decreasing. However, the number of steps needed to\nreach this maximum depends on the effective dimension of the manifold. We attempted to fit our data up to this maximum whenever possible. This amounts to choosing different fit ranges at different regions of the parameter space. The choice of the fit range can be justified by tracking the p-value of the fits.\\\\\n\nAs in previous works \\cite{ambjorn2005spectral,laihoLatticeQuantumGravity2017}, we found the following fit function best represents the data\n\\begin{equation}\n    D_S (\\sigma)=a+\\frac{b}{c+\\sigma}.\n\\end{equation}\nThe fit function yields estimates for the spectral dimension at small distances $D_S(0)$ and also at large distances $D_S(\\infty)$. A single elimination jackknife procedure is used to compute the error-bar, and the fit is performed for different fit ranges. Systematic errors due to the choice of the fit range are added in quadrature with the statistical error of the best fit used to compute the overall error. We use the metric `p-value' to select reasonable fit ranges for the data. Fig.~\\ref{fig_DSinf0_across} shows the variation of $D_S(0)$ and $D_S(\\infty)$  across the transition line from the\ncrumpled to the branched polymer phase, while Fig.~\\ref{fig_DSinf0_along} shows \nthe variation of these quantities along the transition line.\nClearly $D_S$ runs to small ($D_S\\sim 1.5$) values in the UV which is consistent with the earlier EDT studies \\cite{laiho2017recent}, and CDT studies \\cite{coumbeEvidenceAsymptoticSafety2015}. In the IR regime the spectral dimension $D_S(\\infty)$ is larger with $D_S(\\infty)$ varying from $1.82-2.52$. This\nscale dependence of the spectral dimension was also seen earlier in CDT \\cite{ambjorn2005spectral}, renormalization group approach \\cite{lauscher2005fractal}, loop quantum gravity \\cite{modesto2009fractal} and in string theory models \\cite{hovrava2009spectral}. \nIt is not clear from our study whether the UV spectral dimension $D_S(\\infty)$ attains larger values for larger $N_4$. Larger volume simulations with combinatorial triangulations must be conducted to resolve the tension in $D_S(\\infty)$ with the results obtained from the degenerate combinatorial calculations \\cite{laihoEvidenceAsymptoticSafety2011}~\\footnote{In this\nwork we haven't performed a double scaling of this quantity\nusing both the lattice volume and the \nlattice spacing as suggested by Laiho \\textit{et. el.} \\cite{laihoLatticeQuantumGravity2017}. Performing such an\nextrapolation might be important for extracting\na continuum value for the UV spectral dimension.}\n\n\n\n\\section{Conclusions}\nWe have explored the phase diagram of combinatorial Euclidean dynamical\ntriangulation models of four dimensional quantum gravity. Our model\ncontains two parameters - a bare gravitational coupling $\\kappa$\nand a measure parameter $\\beta$.\nWe find evidence for a critical line \n$\\kappa_c(\\beta)$ dividing a crumpled phase from a branched polymer phase in \nagreement with earlier work \\cite{ambjornEuclidian4dQuantum2013,laihoLatticeQuantumGravity2017}. While this line is associated with first order phase\ntransitions for small $\\kappa$ this transition softens with increasing coupling. An intermediate\n``crinkled\" phase opens up in this regime but we have focused our attention on the\nboundary between this region and the branched polymer phase in our analysis since\nthis is the only place where we have observed consistent scaling that survives the\nlarge volume limit. When we refer to the critical point in our results we always mean\nthe boundary between the crinkled and branched polymer phases.\n\nThe focus of much of our work has been\nto compute the Hausdorff and spectral dimensions as we approach this critical\nline from the crumpled phase. We find\nevidence that the Hausdorff dimension $D_H$ along the critical\nline approaches $D_H=4$ as $\\kappa$ increases where it is possible to obtain\nincreasingly good fits to\nto classical de Sitter space. The spectral\ndimension $D_S(s)$ is observed to run with \nscale $s$ attaining values consistent with $D_S(0)=\\frac{3}{2}$ at short distances\nfor all values\nof $\\kappa$. These results are consistent with earlier work using degenerate triangulations\nand causal dynamical triangulation models and models using different measure terms \\cite{ambjornEuclidian4dQuantum2013, laihoLatticeQuantumGravity2017,coumbeEvidenceAsymptoticSafety2015}.\nHowever our measurement of the spectral dimension at long distances\n$D_s(\\infty)$ barely exceeds $D_s(\\infty)\\sim 2$. This result is somewhat in tension with\nthe earlier work. However, \nwe show that $D_s(\\infty)$ depends strongly on the distance in parameter space from\nthe critical line  which renders such measurements delicate and may explain this\ndiscrepancy. Large finite volume effects which have been observed in earlier\nstudies may also make this measurement difficult.\n\n\n\\FloatBarrier\n\n\\acknowledgments \nWe acknowledge Syracuse University HTC Campus Grid and NSF award ACI-1341006\nfor the use of the computing resources. S.C was supported by DOE grant DE-SC0009998.\n\n\\input{Appendix.tex}\n\n\\FloatBarrier\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n Gamma-Ray Bursts (GRBs) are the most energetic transient phenomena in the Universe. The prompt phase of the burst consists of intense gamma-ray flashes, and it can last up to hundreds of seconds in the case of long-duration events. While the exact mechanism for the production of the prompt gamma-ray spectrum is still under debate, it is commonly accepted that the prompt emission is produced within a relativistic collimated plasma outflow launched by the rotating central engine \\citep[for a review see][]{2015PhR...561....1K}. As the plasma propagates in the interstellar medium (ISM) it sweeps up material, causing its gradual deceleration on timescales much longer than the prompt phase duration. This long-lasting emission, which is known as the afterglow, is observed over a wide range of energies (typically from X-rays to radio waves) and it is thought to be produced by synchrotron radiation of relativistic electrons accelerated at the external shock wave \\citep{RM1992,CD1999}. Inverse Compton scattering of low-energy photons by relativistic electrons is typically put forward to explain the recent very high-energy ($E>100$~GeV) photons detections from a handful of GRB afterglows \\citep[for a review see][]{2022Galax..10...66M}.\n\nA very bright GRB was observed on October 9, 2022 by various instruments, including the \\textit{Fermi} Gamma-ray Burst Monitor (GBM) and the Large Area Telescope (LAT) \\citep{GRB221009A_GBM_GCN32642, GRB221009A_LAT_GCN32658}. The Burst Alert Telescope (BAT) of the Neil Gehrels {\\it Swift}\\xspace satellite detected a hard X-ray transient at $T_{\\rm BAT}=59861.59$~MJD, i.e. about an hour later than GBM \\citep{2022ATel15650....1D}.  \nOverall, the prompt emission of GRB~221009A lasts about 330~s \\citep{GRB221009A_GBM_GCN32642}. Preliminary gamma-ray light curves from KONUS-Wind  \\citep{GRB221009A_KONUS_GCN32668} and AGILE \\citep{GRB221009A_AGILE_GCN32650} show a precursor followed by two bright pulses (covering a period of about 100~s), and a fainter pulse starting at $\\sim200$~s after the end of the bright episode. \nObservations of the afterglow with X-shooter at ESO's UT3 of the Very Large Telescope led to the determination of the burst's redshift $z=0.151$~\\citep{2022GCN.32648....1D}. \nMoreover, according to \\citet{2022GCN.32648....1D} multiple spectral features caused by the ISM of the Milky Way were detected, suggesting a large column density of Galactic material along our line of sight. \nThe extreme brightness of this event complicates detailed spectral analysis with instruments like \\textit{Fermi}-GBM and KONUS-Wind due to pile-up effects. Nonetheless, \\citet{GRB221009A_KONUS_GCN32668} estimate the isotropic gamma-ray energy to be $E_{\\rm iso}\\sim 2\\times 10^{54}$~erg using the GBM fluence reported by \\citet{GRB221009A_GBM_GCN32642}. \nThe combination of the proximity to us and the large energy output make this burst an extraordinary event~\\citep[for comparison see Fig.~18 in][]{fermilat2nd}.\nX-ray imaging of the afterglow with \\textit{Swift}-XRT captured several bright rings around the burst's position \\citep{2022ATel15661....1T}. \nThese are formed by scattering of the X-ray burst emission by dust layers in our Galaxy in the direction of the source \\citep[for a recent review on dust scattering and absorption, see][]{2022arXiv220905261C}. \n\nDust scattering rings and halos have been used to study the ISM in the direction of bright X-ray transients with modern observatories \\citep[e.g.][]{2015ApJ...806..265H,2016MNRAS.455.4426V,2016ApJ...825...15H,2016MNRAS.462.1847B,2017MNRAS.468.2532J,2018MNRAS.477.3480J,2019ApJ...875..157J,2021A&A...647A...7L}.\nWhile this is not the first time that dust scattered rings were observed from a GRB \\citep[see e.g.][and references therein]{Klose1994, Vaughan2004, Vianello2007}, the location of GRB~221009A on the sky ($l=52.96^{\\rm o},b=4.32^{\\rm o}$ in Galactic coordinates) and its large inferred isotropic gamma-ray energy offer a unique opportunity to study the Galactic dust via analysis of the ring structures. Here, we analyze publicly available data of \\textit{Swift}-XRT obtained within a few days after the GRB trigger. Our goal is to determine the location of dust layers in the line of sight to the burst by studying the temporal evolution of the dust scattered rings. \n\nThis paper is structured as follows. In Sec.~\\ref{sec:model} we outline the geometrical model used for the description of the X-ray dust rings. In Sec.~\\ref{sec:data} we present the data used for the construction of the angular X-ray surface brightness profiles, and describe the methods applied to the modelling of these profiles. We present our distance measurements in Sec.~\\ref{sec:results}. We continue with a comparison of our results to those obtained from other probes of the dust content in the Galaxy,  and with a discussion on dust grain properties in Sec.~\\ref{sec:discussion}. We finally conclude in Sec.~\\ref{sec:summary} with a summary of our main findings.  \n\n\\section{Modelling of X-ray rings}\\label{sec:model}\nDust is ubiquitous in the interstellar space but the largest dust concentrations (dust layers) are found inside dense cold molecular clouds. X-rays can be preferentially scattered or absorbed (depending on their energy) by interstellar dust grains. In this work we are interested in the geometrical study of the ring structures formed by dust scattering. Therefore we limit our analysis to photon energies $E \\ge 1$~keV. We also neglect multiple X-ray scatterings by dust. \n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{fig1.png} \n\\caption{Schematic illustration (not in scale) of X-ray scattering by dust concentrated in layers located at different distances $d_{\\ell}$ from the satellite. X-ray photons emitted by the GRB, which is located at a distance $d_s \\gg d_\\ell$, travel a distance $\\ell_2$ before changing their direction due to scattering by dust at $d_{\\ell}$. Then, scattered photons travel a distance $\\ell_1$ before reaching the detector. The scattering of X-ray photons by different dust layers that are observed with the same time delay with respect to the burst defines an ellipsoid (red dotted line) with the satellite and the source as its two focal points. The projected image is a smoothed version of the XRT data analysed in this work.}\n\\label{fig:sketch}\n\\end{figure} \n\n The geometrical principles of X-ray scattering by dust layers are illustrated in Fig.~\\ref{fig:sketch}. We consider an X-ray transient occurring at time $t_{\\rm b}$ and at a distance $d_{\\rm s}$. X-ray photons can be scattered in small angles by an intervening dust layer at distance $d_\\ell = x d_{\\rm s}$, where  $x \\ll 1$ for an extragalactic transient (e.g. $x=10^{-5}$ for a transient at 300~Mpc and a dust layer at 3~kpc from us). The scattered photons will be observed with a time delay $\\Delta t$ with respect to the X-ray transient because of their longer path lengths,\n\\begin{equation}\n    \\Delta t =  \\frac{\\ell_1 + \\ell_2}{c} - \\frac{d_s}{c}\n    \\label{eq:delay}\n\\end{equation}\nwhere $\\ell_1 = d_\\ell\/\\cos\\theta = x d_{\\rm }\/\\cos\\theta$, $\\ell_2 = \\sqrt{(d_\\ell \\tan\\theta)^2 + (d_{\\rm s}-d_\\ell)^2} = d_{\\rm s} \\sqrt{(x \\tan\\theta)^2 + (1-x)^2}$, and $\\theta$ is the angular size of the ring (corresponding to the ring radius). For small angles ($\\theta \\ll 1$) the time delay can be approximated (up to second order in $\\theta$) by the following expression\n\\begin{equation}\n    \\Delta t \\approx \\frac{d_{\\rm s}}{2c} \\frac{x}{1-x}  \\theta^2 \\approx \\frac{d_{\\ell}}{2c}\\theta^2. \n\\end{equation}\nPhotons scattered by the same dust layer but arriving with larger time delays will produce a ring of larger angular size. In other words, each ring produced by a single dust layer appears to expand with time. Using the equation above, and assuming $x\\ll 1$, we find an expression for the time evolution of $\\theta$\n\\begin{eqnarray}\n\\theta & \\simeq  & 4.8~{\\rm arcmin} \\left(\\frac{x}{10^{-6}}\\right)^{-1\/2} \\left(\\frac{d_{\\rm s}}{100~\\rm Mpc}\\right)^{-1\/2} \\left(\\frac{\\Delta t}{10^4~\\rm s}\\right)^{1\/2}  \\nonumber  \\\\\n&= &  4.8~{\\rm arcmin} \\left(\\frac{d_{\\ell}}{100~\\rm pc}\\right)^{-1\/2} \\left(\\frac{\\Delta t}{10^4~\\rm s}\\right)^{1\/2}\n\\label{eq:theta}\n\\end{eqnarray}\nThe surface of equal time delays is an ellipsoid with the telescope and the X-ray source placed at the two focal points. Therefore, if multiple dust clouds intersect this surface will produce separate rings of different angular sizes by photons arriving to the observer with the same time delay (see Fig.~\\ref{fig:sketch}). At any given time rings observed with smaller angular sizes are those produced by the more distant layers and vice versa.\n\nThroughout the analysis we adopt a value of $z=0.151$ for the GRB redshift, which corresponds to a luminosity distance $726.5$~Mpc (or a light travel distance $d_{\\rm s}=585.6$~Mpc) based on WMAP9 cosmological parameters \\citep{wmap9}. Eqs.~(\\ref{eq:delay})-(\\ref{eq:theta}) neglect redshift corrections, since the dust scattering layers are located in the Galaxy \\citep[see also][]{1966MNRAS.132..101R,Vaughan2004}.\n\n\\section{Data reduction and analysis}\\label{sec:data}\nWe use data from the Neil\nGehrels {\\it Swift}\\xspace satellite X-ray telescope \\citep[{\\it Swift}\\xspace-XRT,][]{2005SSRv..120..165B}. These were retrieved from the {\\it Swift}\\xspace \\ science data centre\\footnote{\\url{http:\/\/www.swift.ac.uk\/user_objects\/}} and analyzed using standard procedures as outlined in \\citet{2007A&A...469..379E,2009MNRAS.397.1177E}. \nWe use five XRT observations performed between MJD 59862 -- 59866 with obs-id numbers 01126853004, 01126853005, 01126853006, 01126853008 and 01126853009. From the cleaned images we selected events (grade 0-12) with energies between 1 and 10 keV and barycentric corrected times. \n\nOur analysis relies on radial profiles of X-ray photons. Determination of the source's position in the image (i.e. the actual center of the rings) is therefore crucial. Another important effect is the quite rapid expansion of the rings; their angular diameter can evolve significantly on timescales of less than a day -- see Eq.~(\\ref{eq:theta}). We thus split observations into groups of events obtained within a time window of less than 20~ks. We end up with 10 useful subsets of data. We perform source detection and localization in each subset of {\\it Swift}\\xspace-XRT data, and compute the respective exposure maps. Upon correcting each data subset with the appropriate exposure map, we compute radial profiles of X-ray surface brightness (in units of counts s$^{-1}$ arcmin$^{-2}$). \n\n\\begin{figure*} \n\\centering\n\\includegraphics[width=\\textwidth, trim= 0 50 0 0 ]{fig2.jpg}\n\\caption{Time evolution of the angular X-ray surface brightness profile constructed using X-ray photons with $E\\ge1$~keV. For each observation the optimal model (solid red curve), and its decomposition into the various components (dashed blue and orange lines), is overplotted. The peaks of the most prominent (primary) rings identified in the observations are indicated with numbers in each panel. The fourth ring is fitted with two Lorentzians only in the first dataset, since these could not be securely identified in the following datasets. The grey shaded band in each panel indicates the 68 per cent confidence interval.\n}\\label{fig:radialP}\n\\end{figure*}\n\n\\subsection{Modelling of radial profiles}\\label{sec:methods}\nTo model the radial profile of the X-ray surface brightness (in units of counts s$^{-1}$ arcmin$^{-2}$) we use the updated point-source function (PSF) for {\\it Swift}\\xspace-XRT\\footnote{\\url{https:\/\/www.swift.ac.uk\/analysis\/xrt\/pileup.php}}, \n\\begin{equation}\nf_{\\rm PSF}(r) = A \\left [W e^{-\\frac{r^2}{2 \\, \\sigma^2}}+ (1-W) \\left( 1+ \\left(\\frac{r}{r_{\\rm c}}\\right)^2\\right)^{-b}\\right] + B\n\\label{eq:psf}\n\\end{equation}\nwhere $W=0.075$,  $\\sigma =7.42$~arcsec, $r_{\\rm c}=3.72$~arcsec, and $b \\sim 1.31$. In the fitting procedure we leave the power-law index $b$ free to vary and introduce an additional normalization parameter $A$ to account for possible pile up in the detector.  We also add a constant $B$ to account for possible contribution of the background. Each distinctive peak in the angular profile, which corresponds to a ring in the XRT image, is modelled with a Lorentzian function\n\\begin{equation}\nf_{\\rm L}(r) = \\frac{a_{\\rm L}}{\\pi} \\frac{c_{\\rm L}}{(r-b_{\\rm L})^2+c_{\\rm L}^2}\n\\label{eq:lor}\n\\end{equation}\nwhere $a_{\\rm L}$ is the normalization, $b_{\\rm L}$ is the position of the peak, and $2c_{\\rm L}$ is the full width at half maximum. The final fitting function applied to the angular profiles is\n\\begin{equation}\nf_{\\rm tot}(r) = f_{\\rm PSF}(r) + \\sum_{i=1}^{n}f_{\\rm L_{i}}(r)\n\\label{eq:model}\n\\end{equation}\nwhere $n$ is the total number of peaks.  \n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.98\\textwidth ]{fig3.png}\n\\caption{\\textit{Top panel:} Angular profile of the X-ray surface brightness using the stacked X-ray image shown in Fig.~\\ref{fig:im_stack} at a reference time of 2 days since GBM trigger. The optimal model is decomposed into the PSF (dotted line) and multiple Lorentzians  (dashed lines) that are also indicated by Arabic numbers. Roman numbers are used for noting the peaks from the time-resolved angular profiles in Fig.~\\ref{fig:radialP}. The grey shaded region indicates the 68 per cent confidence interval computed from the posterior samples. \n\\textit{Bottom panel:} Plot of residuals computed as the ratio of the difference between the optimal model and the data to the error. }\n\\label{fig:stacked}\n\\end{figure*}\n\\subsection{Analysis of individual datasets}\nTo identify significant peaks in the radial profile distribution we use an iterative process. We start with a radial profile and smooth it with a Savitzky-Golay filter to eliminate noise \\citep{1964AnaCh..36.1627S}. We then identify prominent maxima in the smoothened radial profile (in logarithm) above a certain threshold (i.e. 0.05 in dex) compared to local neighbouring values.\nAs our goal is to identify prominent peaks we are conservative on the choice of the threshold level. In other words, a lower threshold would lead to a few more peaks that would be consistent with noise. \n\nWe then construct a model composed of the PSF and Lorentzian functions -- see Eq.~(\\ref{eq:model}) -- centered at the locations of the identified peaks. We optimize the model to the data (without any smoothing) with a least-square algorithm. We then construct a residual plot with the values normalised over the data uncertainties. Structures in the residual plot can help us identify  secondary peaks. We repeat the procedure to search for secondary peaks in the data above a 3$\\sigma$ level (i.e. 3 times above the errors of each point). This step is crucial since some peaks might be missed initially because they are either very close to other prominent peaks or their peak is hidden by the decay in intensity of the PSF profile, leaving only the side lobes visible. \n\nWe then use the complete model (composed of the PSF and all peaks identified so far) and fit the profiles of each dataset once again using {\\tt emcee} \\citep{2013PASP..125..306F}, a python implementation of the Affine invariant Markov chain Monte Carlo (MCMC) ensemble sampler. This allows us to better estimate the uncertainties in model parameters and to explore possible degeneracies in this multi-parameter problem. \n \nThe iterative procedure described above is applied only to the first dataset with the highest photon statistics. The optimal model is then used as an initial guess for the MCMC sampling of the next dataset. All parameters are sampled from uniform distributions in linear space, except for the background $B$ which is sampled from a uniform distribution in log-space. \nWe produced a chain with 200 walkers that were propagated for 2500 steps each;\nafter testing we concluded that this is an optimal number of steps for the convergence of the walkers. We also discard the first 1000 steps of each chain as burn-in.\n\nWe present in Fig. \\ref{fig:radialP} the angular profiles for 10 individual datasets with the MCMC fitting results overlaid, and list the optimal model parameters for the Lorentzians in Table~\\ref{tab:optimal_params}. In the first angular profile we clearly identify 5 prominent peaks. The fourth ring can be described by two Lorentzian functions. However, we neglect this substructure since these two distinct components are not observed in the following datasets.  As the time progresses the rings are expected to grow apart thus allowing us to to see more structures in the angular profiles, i.e. secondary rings -- see e.g. the bump appearing in the lower panels of Fig.~\\ref{fig:radialP} at smaller angular distances than the first ring. Meanwhile other rings, like the fifth one, can move outside the field of the CCD camera as they expand. It is also possible that some of the dust scattering rings disappear as their intensity faints or due to changes in the ISM properties as each snapshot maps dust scattering at different locations. The spread in the modelled angular profiles becomes larger around peaks at large angular distances where the statistical errors become larger (see e.g. last panel from the left in the top row of Fig.~\\ref{fig:radialP}). This spread is also suggestive of the presence of substructure in the outer rings. A complementary stacking analysis of XRT data, which is presented in the next section, can help us search for such features in the combined angular profile.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{fig4.png}\n\\caption{\nStacked RGB image of all {\\it Swift}\\xspace-XRT observations used in our analysis for a reference time of 2~days post GBM trigger. The image is centered on the GRB location. The position of each photon has been re-scaled from the center of the image (0,0) using the ring expansion law $\\theta \\propto \\sqrt{\\Delta t}$  given in Eq.~(\\ref{eq:theta}). We use the 0.5-10 keV energy range only for this image creation.\n}\n\\label{fig:im_stack}\n\\end{figure}\n\n\\subsection{Stacking analysis of all data}\nAn alternative method to identify dust echoes is to stack all XRT images in order to increase the signal to noise. However, this is not as simple as adding the images because of the dynamic nature of the problem.\nAssuming each and every photon above 1 keV  was scattered once in an intervening dust layer, we can scale its position on the image at an arbitrary time based on the expansion law of Eq.~(\\ref{eq:theta}) and the time the photon was recorded. We define the position of each photon in the image using polar coordinates ($r$, $\\phi$) centered at the location of the GRB. Using the time of arrival of each event we re-scale the $r$ coordinate to $r_{\\rm rs} = r \\, (\\Delta{t_{\\rm rs}}\/\\Delta{t_{\\rm event}})^{1\/2}$, where $\\Delta{t_{\\rm rs}}$ is the reference time for the re-scaled stacked image and $\\Delta{t_{\\rm event}}$ is the time delay between the detection of the photon and the burst. As an indicative example we select $\\Delta{t_{\\rm rs}}=2$~d and use the GBM trigger time as reference time for the GRB, i.e. 13:16:59.99~UT on 09 October 2022 or $T_{\\rm GBM} =59861.55$~MJD~\\citep{GRB221009A_GBM_GCN32642} -- the choice of this reference time will become clearer in the next section. \n\nThe stacking procedure increases the signal to noise in the outer regions, thus enabling us to extend the radial profiles up to a radius of $\\sim25$ arcmin, as illustrated in  Fig. \\ref{fig:stacked}. We also use an adaptive binning for the stacked angular profile with denser sampling for the inner part (i.e. $\\sim$4 versus $\\sim$20~arcsec) for a clearer presentation. After correcting the stacked radial profiles using the individual exposure maps of each snapshot, we follow the same procedure described in the previous section to identify features that could be related to X-ray rings. The analysis of the stacked image, which is shown in Fig.~\\ref{fig:im_stack}, leads to the identification of 16 Lorentzians (see dashed lines in Fig. \\ref{fig:stacked}) that will be discussed further in the following section.\nA model based on Eq.~\\ref{eq:model} was fitted to the radial profiles with a similar procedure as the one described in the previous section, so all parameter quoted are based on the MCMC modelling. \n\n\n\\section{Localization of dust layers}\\label{sec:results} \nWe fit the temporal evolution of the angular radii of the five most prominent rings identified in individual XRT images (Fig.~\\ref{fig:radialP}) using {\\tt emcee} and the expansion law of Eq.~(\\ref{eq:theta}). The statistical uncertainties of the Lorentzian centers (see Table~\\ref{tab:optimal_params}) typically underestimate the uncertainty introduced by our model selection (e.g. PSF with 4 or 5 Lorentzians) and the poor knowledge of priors.  \nThus, when modelling the ring expansion, we add a term $\\ln{f}$ to the likelihood function to account for the systematic scatter and noise not included in the statistical uncertainties of the estimated angular radii \\citep[see similar application][]{2022MNRAS.tmp.3021K}, \n\\begin{equation}\n\\ln{\\mathcal{L}} =  -\\frac{1}{2} \\sum_{i}^{} \\frac{({\\rm model}-{\\rm data})^2}{\\sigma_{\\rm tot, i}^2}+e^{\\sigma_{\\rm tot, i}^2}. \n\\end{equation}\nHere, the total variance is defined as \n\\begin{equation}\n\\sigma_{\\rm tot, \\rm i}^2 = \\sigma_{\\rm i}^2 + e^{2\\ln{f}}, \n\\end{equation}\nwhere $\\sigma_{\\rm i}$ are the errors of the Lorentzian centers $b_{\\rm L, i}$. \n\nOur optimal expansion model for each ring is shown in  Fig.~\\ref{fig:expansion} (see coloured lines), the corner plot with the posterior distributions of all layers is presented in Fig.~\\ref{fig:corner-2} and the dust layer distances are listed in Table~\\ref{tab:params}. The derived time of the burst is $t_{\\rm b}= {\\rm MJD}~59861.53 \\pm 0.02$, which is about one hour and a half earlier than the BAT trigger time $T_{\\rm BAT} =59861.59$~MJD and consistent within errors with the GBM trigger time $T_{\\rm GBM} =59861.55$~MJD~\\citep{GRB221009A_GBM_GCN32642}. \nTherefore, the rings imaged by XRT are produced by X-rays emitted in the prompt phase of the GRB and scattered by dust in our Galaxy. This demonstrates that X-ray photons with energies down to 1 keV are produced during the prompt phase of GRB~221009A, even though they could not be detected by BAT and XRT simultaneously with GBM. Extension of the MeV gamma-ray spectrum to soft X-rays is a common prediction of radiative models, but the prompt X-ray fluence depends on the model details \\citep[see, e.g.,][for lepto-hadronic radiative models of GRB~221009A]{2022arXiv221200766R}. \n\n\nIn regard to the stacked analysis we have demonstrated that by appropriate rescaling of the XRT images we can maintain the information of the peak locations and increase the signal to noise, enabling us to identify more structure in the data. For example several features that appear only in a few snapshots (see Fig.~\\ref{fig:radialP}) are enhanced in the stacked profiles. In Fig.~\\ref{fig:stacked} we can identify at least 8 prominent humps, with one of them  being clearly double peaked (composed of peaks \\#3, \\#4) and some of them being quite broad (i.e. \\#9, \\#10 and \\#12). The angular sizes of all identified peaks and the distances of the corresponding dust scattering locations are summarized in Table~\\ref{tab:params-stacked}. If we consider that the sizes of the rings are just a projection effect, we need to use the estimated distances in order to ascertain if two nearby rings may be associated with the same dust layer and appear as separate due to inhomogeneities in the dust distribution of a single cloud. In fact the four innermost rings that appear to overlap the most in the angular profile are those that are physically the most detached, since the relevant dust layers are located at distances of about 14.7 kpc, 9.07 kpc, 4.4 kpc and 3.4 kpc. Thus, they cannot be associated with the same production site. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47\\textwidth]{fig5.jpg}\n\\caption{Temporal evolution of the angular size of rings detected in individual {\\it Swift}\\xspace-XRT images (coloured symbols). \nThe most probable model for each ring is overplotted with a dashed line of the same colour as the markers. The grey shaded regions indicate the 99.7 per cent confidence intervals. Black markers indicate the rings identified in the stacked image at a reference time of 2 days post GBM trigger. Dotted grey curves show the predicted expansion according to Eq.~(\\ref{eq:theta}).}\n\\label{fig:expansion}\n\\end{figure}\n\n\nThe innermost peak of the stacked data is also seen in individual snapshots (see e.g. the last two panels in the bottom row of Fig.~\\ref{fig:radialP}), but its structure does not remind that of an extended halo. To check if these innermost peaks follow the $\\sqrt{\\Delta} t$ expansion law, we performed an additional fit to the last 4 individual datasets by adding two more Lorentzian functions. However, the Lorentzian centers do not seem to follow the expansion law.\nGiven that our results are limited by the {\\it Swift}\\xspace\/XRT angular resolution, the origin of these features should be revisited with follow-up analysis of \\hbox{\\it Chandra}\\xspace data.\n\n\\renewcommand*{\\arraystretch}{1.1} \n\\begin{table}\n\\centering \n\\caption{Dust layer distances obtained from fitting the expansion of the most prominent rings in individual images using Eq.~(\\ref{eq:theta}).} \n\\label{tab:params}\n\\begin{threeparttable}\n\\begin{tabular}{c c c}\n\\toprule\nRing  & $x$ & $d_\\ell$~(kpc) \\\\\n\\hline \nI  & $(6.23 \\pm 0.11)\\times 10^{-6}$ &${3.65 \\pm 0.06}$ \\\\\nII  & ${(3.32 \\pm 0.05)}\\times 10^{-6}$ & ${1.95 \\pm 0.03}$ \\\\\nIII  &  ${(1.18 \\pm 0.02)}\\times 10^{-6}$ & ${0.691 \\pm 0.008}$ \\\\\nIV  & ${(7.45 \\pm 0.09)}\\times 10^{-7}$ & ${0.436 \\pm 0.005}$ \\\\\nV  & ${(2.85 \\pm 0.04)}\\times 10^{-7}$ & ${0.167 \\pm 0.003}$\\\\\n\\bottomrule\n\\end{tabular}\n\\begin{tablenotes}\n\\item {Note. --} The listed values and errors correspond to the median value and the 68 per cent range of the posterior distributions, respectively. \n\\end{tablenotes}\n\\end{threeparttable}\n\\end{table}\n\n\\begin{table}\n\\centering \n\\caption{Locations of dust scattering regions obtained from fitting all peaks in the angular profile of the stacked X-ray image.} \n\\label{tab:params-stacked}\n\\begin{threeparttable}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{tabular}{p{0.2\\columnwidth}p{0.3\\columnwidth}p{0.3\\columnwidth}}\n\\toprule\nRing\\tnote{*} & $\\theta$ (arcmin) & $d_\\ell$~(kpc)\\\\\n\\hline\n1 & 1.644$_{- 0.03 }^{+ 0.011 }$& 14.7$_{- 0.2 }^{+ 0.5 }$\\\\ \n2 & 2.095 $_{- 0.02 }^{ 0.011 }$& 9.07$_{- 0.09 }^{+ 0.2 }$\\\\\n3 (I) & 3.009$_{- 0.012 }^{+ 0.008 }$& 4.40$_{- 0.02}^{+ 0.03}$\\\\\n4 (I) & 3.402$_{- 0.013 }^{+ 0.005 }$& 3.440$_{- 0.011 }^{+ 0.03 }$\\\\ \n5 (II) & 4.505$_{- 0.011 }^{+ 0.009 }$& 1.961$_{- 0.008 }^{+ 0.009 }$\\\\ \n6 & 5.804$_{- 0.03 }^{+ 0.020 }$& 1.182$_{- 0.007 }^{+ 0.012 }$\\\\ \n7 & 6.209$_{- 0.009 }^{+ 0.020 }$& 1.033$_{- 0.007}^{+ 0.003 }$\\\\ \n8 (III) & 7.624$_{- 0.008 }^{+ 0.008 }$& 0.6849$_{- 0.0014 }^{+ 0.0014 }$\\\\ \n9 (IV) & 9.538$_{- 0.03 }^{+ 0.011 }$& 0.4376$_{-0.0011 }^{+ 0.002 }$\\\\ \n10 (IV) & 10.19$_{- 0.03 }^{+ 0.02 }$& 0.3835$_{-0.0008 }^{+ 0.002}$\\\\ \n11 & 12.02$_{- 0.02 }^{+ 0.03 }$& 0.2753$_{- 0.0013}^{+ 0.0011 }$\\\\\n12 (V) & 15.43$_{- 0.03 }^{+ 0.02 }$& 0.1673$_{- 0.0004}^{+ 0.0006}$\\\\ \n13 & 17.14$_{- 0.02 }^{+ 0.03 }$& 0.1356$_{- 0.0004 }^{+ 0.0003 }$\\\\\n14 & 18.71$_{- 0.02 }^{+ 0.02 }$& 0.1138$_{- 0.0002 }^{+ 0.0002 }$\\\\ \n15 & 21.41$_{- 0.02 }^{+ 0.04 }$& 0.0869$_{- 0.0003 }^{+ 0.00014 }$\\\\ \n16 & 23.14$_{- 0.02 }^{+ 0.02 }$& 0.07434$_{- 0.00014 }^{+ 0.00006 }$\\\\\n\\bottomrule\n\\end{tabular}\n\\renewcommand{\\arraystretch}{1}\n\\begin{tablenotes} \n\\item[*] The numbers enclosed in parentheses correspond to the five rings presented in Table~\\ref{tab:params}. \n\\item {Note. -- } The listed values and errors correspond to the median value and the 68 per cent range of the posterior distributions, respectively. \n\\end{tablenotes}\n\\end{threeparttable}\n\\end{table}\n\nAnother interesting feature is seen at the residual plot of Fig.~\\ref{fig:stacked} close to the locations of peaks \\#8, \\#9 and \\#10. First, the residual structure around \\#8 indicates multiple peaks that are not resolved.  Second, large residuals are found before and after the peaks \\#8 and \\#10 respectively. These residuals are caused by the width of Lorentzian profiles used for describing peaks \\#8 and \\#10 that lead to excess emission over the data. Clearly the mathematical description could be improved by inserting two more Lorentzian lines. Higher resolution instruments like \\textit{Chandra} could potentially identify more peaks in this range of angles that would correspond to layer distances between 0.4-0.7 kpc. We finally note that the outermost rings translate to layers at distances of only 74~pc. This is intriguing and highlights the power of X-ray tomography in providing distance measurements to dust layers even in regions of the Galaxy that cannot be mapped as accurately by other techniques. \n\n\\begin{figure}\n\\includegraphics[width=0.49\\textwidth, trim = 0 75 0 0]{fig6.png}\n\\caption{Comparison of dust layer distances with various means of Galactic extinction measures in the direction of GRB~221007A.  \\textit{Top panel:} Extinction profile (in terms of the monochromatic extinction at 5500 \\AA) based on Gaia EDR3 and 2MASS data \\citep{2022A&A...661A.147L}.  \\textit{Middle panel:} Posterior distribution of reddening based on {\\tt Bayestar19} 3D extinction maps \\citep{2015ApJ...810...25G,2019ApJ...887...93G}. \\textit{Bottom panel}: Posterior distribution of the mean extinction at 5495 \\AA \\, based on IPHAS photometry \\citep{2014MNRAS.443.2907S}.  The reliability range of the extinction estimates is indicated with grey shaded areas. Vertical lines indicate the location of the dust layers found in the stacked data (top and middle panels) or in the individual data (bottom panel). } \\label{fig:ebv}\n\\end{figure}\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\\subsection{Comparison with other probes of dust}\nThe dust content in our Galaxy is typically studied via reddening of starlight and CO emission from cold gas, while dust-scattering rings offer a new dimension to the above.\nAfter estimating the location of the dust layers we can compare their position with the Galactic extinction profile along the line of sight due to dust attenuation. We first use the data from {\\tt Bayestar19} 3D maps, i.e. the latest version of the Dust Map based on Gaia, Pan-STARRS 1, and 2MASS data \\citep{2015ApJ...810...25G,2019ApJ...887...93G}. Given the probabilistic nature of the maps we extract 1000 random samples for the direction of our source and estimate the median and 68 per cent confidence range for the differential reddening value.  We note that the output values of the 3D map are given in arbitrary units; we refer the reader to \\cite{2015ApJ...810...25G,2019ApJ...887...93G} for a description of the conversion to $E(B-V)$ or extinction $A$ in a specific pass band. We also extract the mean extinction (at the reference wavelength of 5495 \\AA) along the direction of the burst from \\cite{2014MNRAS.443.2907S} who derived the 3D map of extinction in the northern Galactic plane ($|b|<5^{\\rm o}$) using IPHAS DR2 photometry.\nThe IPHAS map provides cumulative extinction values which for the direction of the system correspond to about 4 magnitudes (up to a distance of $\\sim 6$~kpc where the results are trustworthy). The extinction can also be used as a proxy for hydrogen column density according to $N_{\\rm H}=2.21\\times10^{21}$~$A_{\\rm V}$~cm$^{-2}$ assuming solar metallicity \\citep{2009MNRAS.400.2050G}. The estimated column density is $N_{\\rm H} \\sim 0.9 \\times 10^{22}$~cm$^{-2}$ (assuming $A_0\\approx A_{\\rm V}$). Both extinction maps discussed so far have low resolution to smaller distances (within 1 kpc). Therefore, to obtain a better picture of the local extinction profile we use the updated Gaia-2MASS 3D maps of Galactic interstellar dust \\citep{2022A&A...661A.147L}, which are available via the {\\tt G-TOMO} online tool in the EXPLORE website\\footnote{\\url{https:\/\/explore-platform.eu}}.  \n\nThe results are shown in Fig.~\\ref{fig:ebv} where the vertical lines indicate the locations of the dust layers derived from the analysis of individual XRT datasets (bottom panel) and of the stacked image (top and middle panels). There is some agreement between the inferred distances for the nearby layers ($\\lesssim$ 1~kpc) and the positions of larger $A_{\\rm 0}$ (and thus $N_{\\rm H}$) values. Estimates for the amount of dust from extinction measurements are limited to smaller distances, since the amount of stars and the accuracy of photometry decreases as we move to the outskirts of the Galaxy. For instance, the extinction estimates from IPHAS are not trustworthy beyond $\\sim 6~$kpc (see shaded regions in panels of Fig.~\\ref{fig:ebv}). Meanwhile, X-ray scattering by dust closer to us produces rings with larger angular sizes that are more difficult to detect due to e.g. lower intensity. Overall, performing an X-ray tomography of the Galaxy via dust scattering echoes favours the detection of layers at larger distances (the scattering angle is smaller and the ring intensity larger), thus complementing photometric techniques for dust mapping. \n\n\\begin{figure}\n\\setbox1=\\hbox{\\includegraphics[width=0.48\\textwidth]{fig7.png}}\n\\includegraphics[width=0.48\\textwidth]{fig7.png}\\llap{\\raisebox{4.3cm}{\\includegraphics[height=3.3cm]{fig8.png}}}\n\\caption{\nVelocity-integrated spatial CO sky-map with brighter colours corresponding to larger values \\citep{2001ApJ...547..792D}. Inset plot shows a zoom-in version of the central image. A circle with angular size of 12\\arcmin~ marks the location of GRB 221009A, its size is comparable to the radius of the observed rings, the size is also comparable to the CO map resolution (i.e. pixel size).\n}\n\\label{fig:co}\n\\end{figure}\n\n\n\\begin{figure*}\n\\includegraphics[width=0.99\\textwidth]{fig9.png}\n\\vspace{-1cm}\n\\caption{Location of massive molecular clouds (blue circles) in the Galactic plane as obtained from CO measurements \\citep{2016ApJ...822...52R}. The size of the markers corresponds to the actual cloud size. The direction of GRB 221007A is marked with a magenta dashed line. The most prominent dust layers at distances of $\\sim$1.03, 1.18, 1.96, 3.44, 4.40, 9.07 and 14.7 kpc are marked with magenta circles,  and Arabic numbers corresponding to the rings 7, 6, 5, 4, 3, 2, 1 respectively (see Table \\ref{tab:params-stacked}). The Sun's location is marked with a yellow circle. Background illustration of the Milky Way reflecting the Galactic structure [Image credit: NASA\/JPL-Caltech\/ESO\/R. Hurt] }\n\\label{fig:MW}\n\\end{figure*}\n\nTo better visualize the direction of the source compared to the Galactic plane we show in Fig.~\\ref{fig:co} its location in the sky on top of the velocity-integrated spatial CO map \\citep{2001ApJ...547..792D}. \nThe map provides radial velocities that could be de-projected and translated to distances. However, this is far from an easy task, which does not always result in a unique solution for the distance of the CO emitting gas, but can yield instead a near and a far distance solution.  \\citet{2016ApJ...822...52R} used a dendrogram-based decomposition of the \\cite{2001ApJ...547..792D} survey and constructed a catalog of 1064 massive molecular clouds\nthroughout the Galactic plane. These massive cold clouds are another tracer of dust concentrations in our Galaxy. In Fig. \\ref{fig:MW} we project the catalog of the molecular clouds  (blue points) onto an illustration of the Milky way and compare those with the dust layers as inferred from the rings at distances of $\\sim$1.03, 1.18, 1.96, 3.44, 4.40, 9.07 and 14.7 kpc (magenta points).\nWe did not identify any dust layers between 5 and 9 kpc through the ring analysis, which agrees with the paucity in the molecular cloud distribution and the gap between the Sagittarius and Perseus spiral arms (Fig.~\\ref{fig:MW}). We note that the molecular clouds are confined to the Galactic plane ($|b| \\lesssim 2^{\\rm o}$) with radii of the order of 100~pc,  \nwhile our line of sight probes dust distributed above the plane ($b=4.32^{\\rm o}$).  Even though a direct connection of the cloud and layer distributions cannot be made, it is plausible that the dust extending above the plane follows a similar distribution as the one probed by the clouds.  \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig10.png}\n\\caption{Scattered flux of X-ray rings  (in arbitrary units) integrated over the angular extent and plotted as a function of the angular size $\\theta$ (coloured symbols). Theoretical expectations based on the simplest grain model are overplotted for indicative parameter values: $a_{\\max} = 0.3~{\\mu m}, a_{\\min}=a_{\\max}\/10$ (solid lines), $a_{\\max} = 0.1~{\\mu m}, a_{\\min}=a_{\\max}\/10$ (dashed lines), and $a_{\\max} = 0.3~{\\mu m}, a_{\\min}=a_{\\max}\/3$ (dotted lines). In all cases, $q=4$. Theoretical curves (for each parameter set) are normalized to the same value at 1~arcmin.\n}\n\\label{fig:Fsc}\n\\end{figure}\n\n\\subsection{Scattered X-ray intensity}\nThe evolution of the X-ray scattered intensity with time (or angular size) is associated with the dust grain properties.\nThe X-ray flux of a ring with angular size $\\theta$, which is produced by scattering of a infinitesimally short duration burst of X-rays with fluence $S_{\\rm X}(E)$ by dust at distance $x_{\\rm i}$, can be written as \\citep[for details see][]{2016MNRAS.455.4426V} \n\\begin{equation}\nF_{\\rm sc}(E,\\theta) = \\frac{C_{\\rm i} N_{\\rm d, i} S_{\\rm X}(E) }{x_{\\rm i}(1-x_{\\rm i}) }\\int_{\\tilde{a}_{\\min}}^{\\tilde{a}_{\\max}} {\\rm d}\\tilde{a} \\, \\tilde{a}^{6-q}\\exp\\left(-\\frac{\\theta^2}{2 (1-x_{\\rm i})^2 \\Theta^2(E,\\tilde{a})}\\right)\n\\label{eq:Fsc}\n\\end{equation}\nwhere $C_{\\rm i}$ is a normalization constant that depends on the metallicity and mass density of dust in layer $i$ and is of order unity for typical parameters \\citep[e.g.][]{2016MNRAS.455.4426V} and $N_{\\rm d, i}$ is the dust column density of the $i$-th layer. The integral of the differential scattering cross section, which is modelled using the Rayleigh-Gans approximation~\\citep[e.g.][]{Mauche1986}, is performed over a power-law grain size distribution with slope $q$~\\citep[][]{MRN1977}; here $\\tilde{a}$ is the grain size in $\\mu m$ and $\\Theta$ is the typical angular size of a ring produced via scattering of 1~keV photons on grains with radius 0.1~$\\mu m$, \n\\begin{equation}\n\\Theta = 10.4 \\, {\\rm arcmin} \\left(\\frac{1~\\rm keV}{E} \\right) \\left( \\frac{0.1~\\rm \\mu m}{a}\\right).\n\\label{eq:theta2}\n\\end{equation} \nFor photon energies $E > 1$~keV, as those considered in this paper, Eq.~(\\ref{eq:Fsc}) is valid for $\\tilde{a} \\ll 1$. Most photons in the analyzed XRT images have energies between 1 and 2 keV. We therefore integrate the flux given by Eq.~(\\ref{eq:Fsc}) over this narrow band and perform a qualitative comparison to the scattered fluxes derived from the optimal angular-profile models of the rings (see Fig.~\\ref{fig:radialP}). We model the prompt X-ray fluence as $S(E) \\propto (E\/E_{\\rm pk})^{-\\Gamma+1}$, where $E_{\\rm pk}=1060$~keV is the observed peak energy of the prompt spectrum as estimated from KONUS-WIND \\citep{GRB221009A_KONUS_GCN32668} and $\\Gamma = 3\/2$ is the photon index of the prompt GRB spectrum, assuming a fast-cooling synchrotron spectrum extending down to 1~keV \\citep{2022arXiv221200766R}. \n\nThe theoretical expectations for indicative dust parameters \nare shown in Fig.~\\ref{fig:Fsc}. In all cases, we assume a power-law size distribution of grains with slope $q=4$ extending from $a_{\\min}$ to $a_{\\max}$. Solid lines correspond to $a_{\\max}=0.3 \\mu m, a_{\\min}=a_{\\max}\/10$, dashed lines to $a_{\\max}=0.1 \\mu m, a_{\\min}=a_{\\max}\/10$, and dotted lines to $a_{\\max}=0.3 \\mu m, a_{\\min}=a_{\\max}\/3$. We do not determine the normalization parameter for each dust layer, $\\tilde{C}_{\\rm i}=C_{\\rm i}N_{\\rm d,i}$, as we are interested in the relative ratio of the fluxes. Even without fitting the model to the data we can draw some useful conclusions. First, the maximum grain size cannot be much smaller than $0.3~\\mu m$. For example, $F_{\\rm sc}(\\theta)$ would be almost constant for $\\theta \\lesssim 10$~arcmin if $a_{\\max}=0.1~\\mu m$ in contradiction to the data (see dashed lines). The smooth turnover of $F_{\\rm sc}(\\theta)$ is related to the exponential cutoff in the scattering cross section (see Eq.~(\\ref{eq:Fsc})), and occurs approximately at $\\Theta(\\bar{E}, a_{\\max})$, which is $\\simeq 2.5$~arcmin for a mean photon energy $\\bar{E}=1.5$~keV and $a_{\\max}=0.3~\\mu m$ -- see Eq.~(\\ref{eq:theta2}). Second, the minimum grain size cannot be easily constrained because of the small dynamic range of the ring angular sizes. In general, the scattered X-ray flux follows a power law in angle, with a slope depending on $q$, and an extent determined roughly by $\\Theta(\\bar{E}, a_{\\max})$ and $\\Theta(\\bar{E}, a_{\\min})$ -- see e.g. green and red solid lines. As $a_{\\min}$ approaches $a_{\\max}$, however, the power-law segment of $F_{\\rm sc}(\\theta)$ becomes shorter, till the point that we start seeing the exponential cutoff of the scattering cross section for grains of typical size $a_{\\min}\\sim a_{\\max}$ (compare solid and dotted lines). Grain distributions with $a_{\\min} \\ll a_{\\max}$ or $a_{\\min} \\sim a_{\\max}$ are compatible with the data for rings I, II, and V. In fact, the scattered flux of the fifth ring would be better described by a model of grains with similar size instead of an extended power-law distribution (compare purple solid and dotted lines). \nThird, grain distributions with $q\\sim 3.5-4$ and $a_{\\min} \\ll a_{\\max}$ can produce the observed power-law decline of the scattered flux with angular size for rings III and V. Lastly, the relative normalizations for the dust layers are $\\tilde{C}_{\\rm I}:\\tilde{C}_{\\rm II}:\\tilde{C}_{\\rm III}:\\tilde{C}_{\\rm IV}:\\tilde{C}_{\\rm V} = 1:0.15:0.9:1.4:0.3$. The relative normalizations can be used to order the dust scattering production sites in terms of increasing optical depth or amount of dust contained in each layer, with the fourth layer (at 0.44~kpc) being the one with the largest dust content.\n\nPrompt X-ray scattering by dust in the GRB host galaxy can also be imprinted in the X-ray afterglow emission \\citep[e.g.][]{1998ApJ...507..300K, 2007ApJ...660.1319S}. For instance, the strong hard-to-soft evolution of the X-ray emission observed in the afterglow of the ultra-long GRB 130925A could be explained by this phenomenon \\cite{2014MNRAS.444..250E}. The X-ray echoes of GRB 221009A are instead produced via scattering of prompt X-ray photons by dust in our Galaxy, as demonstrated in Sec.~\\ref{sec:results}. Still, spectral softening with time is also expected. However, the X-ray afterglow of GRB 221009A shows no evidence for strong spectral evolution with a photon index close to -2 for about two decades in time\\footnote{ \\url{https:\/\/www.swift.ac.uk\/burst_analyser\/01126853\/}}. In the small-angle scattering approximation, the scattered flux shows a shallow decline with time, i.e. $t^{-1\/4}$ -- see e.g.~ Eq.~(3) in \\cite{2007ApJ...660.1319S}. A steeper decline approaching $t^{-2}$ is expected after $t \\gtrsim 1.6\\times 10^5~{\\rm s} \\left(E\/{1~{\\rm keV}}\\right)^{-2}\\left(a\/0.1~\\mu m\\right)^{-2}\\left(d_\\ell\/100~{\\rm pc}\\right)(1+z_{\\rm s})$. Therefore, a transition from a shallow decay to a steeper decline in the X-ray scattered flux would be expected somewhere between $6.5\\times10^4$~s and $\\sim 1.5\\times10^6$~s  for layers at distances between 0.4~kpc and 9.6~kpc, respectively.  The XRT light curve shows no evidence of such transition, and its flux decays almost as a single power law (with slope $\\sim -1.6$ for $t\\gtrsim 10^4$~s after the GBM trigger. Comparison of dust-scattering models to the XRT afterglow light curve might help to constrain the dust column density of each layer and estimate the contribution of the scattered flux to the intrinsic non-thermal emission from the GRB blast wave. \n\n \n\\section{Conclusions}\\label{sec:summary}\n\nIn this paper we have analyzed publicly available {\\it Swift}\\xspace-XRT data that were obtained within a few days after the detection of GRB~221009A. We constructed angular profiles of photons with energies above 1 keV from individual XRT images, and identified the most prominent peaks. By modelling their temporal evolution over a course of several days we were able to determine the time of the X-ray burst and the distances of five intervening dust layers. Complementary analysis of the stacked XRT image (scaled to a reference time of two days after the burst) revealed a richer angular structure with 16 peaks due to the increased photon statistics.  The main conclusions of our work are the following:\n\\begin{itemize}\n    \\item The expansion of the five more prominent peaks in the time-resolved angular profiles yields the time of the X-ray burst, which is consistent with the GBM trigger (i.e. the prompt X-ray spectrum should extend to 1~keV). \n    \\item Analysis of the stacked image reveals extra features and increases the number of potential dust concentrations along the line of sight to at least 16, spanning from 0.07~kpc to 15 kpc. This is this the largest distance range probed by X-ray scattering echoes so far.\n    \\item Locations of dust layers are generally consistent with local maxima of the radial extinction profile, while the absence of dust layers between 5 and 9 kpc coincides with the gap between the Sagittarius and Perseus spiral arms. \n    \\item The evolution of the scattered X-ray flux (for the five more prominent rings) with angular size is consistent with scattering by dust grains having a power-law size distribution with slope $q\\sim 3.5-4$ and maximum grain size of $0.3 \\mu m$. For the closest layer to us, the minimum grain size could be comparable to $a_{\\max}$. \n\\end{itemize}\n\n\\section*{Acknowledgements}\nWe thank the referee for comments that helped to improve the manuscript. We are also grateful to Dr. Andrea Tiengo for identifying a typo in Eq.~(3) and for useful discussions. The authors acknowledge support by H.F.R.I. through the project ASTRAPE (Project ID 7802) and the project UNTRAPHOB (Project ID 3013). M.P. also acknowledges support from the MERAC Fondation through the project THRILL.\n\n\\section*{Data availability}\nX-ray data are available through the High Energy Astrophysics Science Archive Research Center: \\url{heasarc.gsfc.nasa.gov}. \nThe python notebooks used for the X-ray image analysis and the radial profile fitting will be made available upon reasonable request to the authors.\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\u306f\u3058\u3081\u306b}\n\\subsection{\u8a18\u53f7}\n\u672c\u7bc0\u3067\u306f\u672c\u8ad6\u6587\u5168\u4f53\u3067\u5171\u901a\u3057\u3066\u7528\u3044\u308b\u8a18\u53f7\u306e\u5b9a\u7fa9\u3092\u8ff0\u3079\u308b. \n\n$\\mathbb{N} = \\{1,2,3,\\ldots\\}$\u3092\u81ea\u7136\u6570\u5168\u4f53\u306e\u96c6\u5408\u3068\u3057, $\\mathbb{Z}$\u3092\u6574\u6570\u5168\u4f53\u306e\u96c6\u5408\u3068\u3057, $\\mathbb{Q}$\u3092\u6709\u7406\u6570\u5168\u4f53\u306e\u96c6\u5408\u3068\u3059\u308b.  \u307e\u305f$\\mathbb{R}$\u3067\u5b9f\u6570\u5168\u4f53\u306e\u96c6\u5408\u3092\u8868\u3059. \n\u7279\u306b\u65ad\u3089\u306a\u3051\u308c\u3070, $\\mathbb{R}^n$\u306f\u901a\u5e38\u306e\u8ddd\u96e2\u306b\u3088\u3063\u3066\u4f4d\u76f8\u7a7a\u9593\u3068\u307f\u306a\u3059. \n\\begin{defn}\n  \u4f4d\u76f8\u7a7a\u9593$X,Y$\u306b\u5bfe\u3057\u3066, \n  \\[\n  C(X,Y) := \\{f:X \\rightarrow Y \\mid f\\mbox{\u306f\u9023\u7d9a}\\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. $C(X,\\mathbb{R})$\u306f\u5358\u306b$C(X)$\u3068\u8868\u3059. \u307e\u305f, \u95a2\u6570$f:X \\rightarrow \\mathbb{R}$\u306b\u5bfe\u3057\u3066, \n \\[\n \\mathrm{supp}(f) := \\mathrm{cl}(\\{x \\in X \\mid f(x) \\neq 0 \\})\n \\]\n \u3068\u5b9a\u7fa9\u3057, \u958b\u96c6\u5408$\\Omega \\subset \\mathbb{R}^n$\u3068$m = 0,1,2,\\ldots$\u306b\u5bfe\u3057\u3066\n \\[\n \\begin{aligned}\n &C_0(X) := \\{ f \\in C(X) \\mid \\mathrm{supp}(f) \\mbox{\u306f\u30b3\u30f3\u30d1\u30af\u30c8}\\} \\\\\n &C^m(\\Omega) := \\{f \\in C(\\Omega) \\mid f\\mbox{\u306f}\\Omega \\mbox{\u4e0a\u3067}m\\mbox{\u968e\u504f\u5fae\u5206\u53ef\u80fd\u3067\u5168\u3066\u306e\u504f\u5c0e\u95a2\u6570\u304c\u9023\u7d9a}\\} \\\\\n &C^\\infty(\\Omega) := \\bigcap_{m=0}^{\\infty} C^m(\\Omega)\n \\end{aligned}\n \\]\n \u3068\u5b9a\u7fa9\u3059\u308b. \n\\end{defn}\n\\begin{defn}[\u751f\u6210\u3055\u308c\u308b$\\sigma$-\u52a0\u6cd5\u65cf] \\\\\n  $X$\u3092\u96c6\u5408\u3068\u3057, $\\mathcal{A}_0 \\subset \\mathcal{P}(X)$\u3068\u3059\u308b($\\mathcal{P}(X)$\u306f$X$\u306e\u90e8\u5206\u96c6\u5408\u5168\u4f53\u306e\u96c6\u5408). \u3053\u306e\u3068\u304d, $\\mathcal{A}_0$\u3067\u751f\u6210\u3055\u308c\u308b$X$\u4e0a\u306e$\\sigma$-\u52a0\u6cd5\u65cf$\\sigma[\\mathcal{A}_0]$\u3092\n  \\[\n  \\sigma[\\mathcal{A}_0] := \\bigcap \\{ \\mathcal{A} \\mid \\mathcal{A} \\supset \\mathcal{A}_0, \\mathcal{A}\\mbox{\u306f}X\\mbox{\u4e0a\u306e}\\sigma\\mbox{-\u52a0\u6cd5\u65cf} \\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \n\\end{defn}\n\\begin{defn}[Borel $\\sigma$-\u52a0\u6cd5\u65cf, Borel\u6e2c\u5ea6] \\\\\n  $(X,\\mathcal{O})$\u3092\u4f4d\u76f8\u7a7a\u9593\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $X$\u4e0a\u306eBorel $\\sigma$-\u52a0\u6cd5\u65cf$\\mathcal{B}_X$\u3092$\\mathcal{B}_X = \\sigma[\\mathcal{O}]$\u3068\u5b9a\u7fa9\u3059\u308b. \u305d\u3057\u3066, $X$\u4e0a\u306eBorel\u6e2c\u5ea6\u3068\u306f, \u53ef\u6e2c\u7a7a\u9593$(X,\\mathcal{B}_X)$\u4e0a\u306e\u6e2c\u5ea6\u306e\u3053\u3068\u3092\u3044\u3046.  $\\mathbb{R}^n$\u306f$(\\mathbb{R}^n,\\mathcal{B}_{\\mathbb{R}^n})$\u3068\u3057\u3066\u53ef\u6e2c\u7a7a\u9593\u3068\u307f\u306a\u3059. \n\\end{defn}\n\\begin{defn}[\u53ef\u6e2c\u5199\u50cf, $L^p$\u7a7a\u9593] \\\\\n  $(X,\\mathcal{M}),(Y,\\mathcal{N})$\u3092\u53ef\u6e2c\u7a7a\u9593\u3068\u3059\u308b.\n  \u3053\u306e\u3068\u304d, $X$\u304b\u3089$Y$\u3078\u306e\u53ef\u6e2c\u5199\u50cf\u5168\u4f53\u306e\u96c6\u5408\u3092$\\mathbb{M}(X,Y)$\u3067\u8868\u3059: \n  \\[\n  \\mathbb{M}(X,Y) := \\{f:X \\rightarrow Y \\mid \\forall E \\in \\mathcal{N}, f^{-1}(E) \\in \\mathcal{M}\\}\n  \\]\n  \u307e\u305f, $(X,\\mathcal{M})$\u4e0a\u306e\u6e2c\u5ea6$\\mu$\u3068$1 \\leq p < \\infty$\u306b\u5bfe\u3057\u3066, \n  \\[\n  L^p(X,\\mathcal{M},\\mu) := \\{f \\in \\mathbb{M}(X,\\mathbb{R}) \\mid \\int_{X} |f|^p d\\mu < \\infty\\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3057, $f \\in L^p(X,\\mathcal{M},\\mu)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\lVert f \\rVert_{L^p} = \\left( \\int_{X} |f|^p d\\mu \\right)^{1\/p}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u6587\u8108\u304b\u3089\u63a8\u5bdf\u3067\u304d\u308b\u5834\u5408\u306b\u306f$L^p(X,\\mathcal{M},\\mu)$\u3092\u5358\u306b$L^p(X)$\u3068\u8868\u3059.  \u3055\u3089\u306b, \n  \\[\n  L^{\\infty}(X,\\mathcal{M},\\mu) := \\{f \\in \\mathbb{M}(X, \\mathbb{R}) \\mid \\mu\\mbox{-a.e.\u3067\u6709\u754c}\\}\n  \\]\n  \u3068\u304a\u304d, $f \\in L^{\\infty}(X,\\mathcal{M},\\mu)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\lVert f \\rVert_{L^{\\infty}} = \\inf \\{\\lambda \\geq 0 \\mid \\mu(|f|\\geq \\lambda) = 0\\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u6587\u8108\u304b\u3089\u63a8\u5bdf\u3067\u304d\u308b\u5834\u5408\u306b\u306f$L^\\infty(X,\\mathcal{M},\\mu)$\u3092\u5358\u306b$L^{\\infty}(X)$\u3068\u8868\u3059.\n\\end{defn}\n\u306a\u304a, $L^p(X,\\mathcal{M},\\mu)$\u306b\u5c5e\u3059\u308b\u95a2\u6570\u306fa.e.\u3067\u4e00\u81f4\u3059\u308b\u3068\u304d\u7b49\u3057\u3044\u3068\u307f\u306a\u3059. \u3064\u307e\u308a, $f,g \\in L^p(X,\\mathcal{M},\\mu)$\u306b\u5bfe\u3057\u3066, $f=g$\u3068\u306f$\\mu(\\{x \\in X \\mid f(x) \\neq g(x)\\}) = 0$\u3068\u306a\u308b\u3053\u3068\u3092\u3044\u3046. \n\\begin{defn}[\u30a2\u30d5\u30a3\u30f3\u95a2\u6570]\n $r \\in \\mathbb{N}$\u306b\u5bfe\u3057\u3066, \n \\[\n \\mathbb{A}^r := \\{f:\\mathbb{R}^r \\rightarrow \\mathbb{R} \\mid f\\mbox{\u306f\u30a2\u30d5\u30a3\u30f3\u95a2\u6570}\\}\n \\]\n \u3068\u304a\u304f. \u305f\u3060\u3057, $f:\\mathbb{R}^r \\rightarrow \\mathbb{R}$\u304c\u30a2\u30d5\u30a3\u30f3\u95a2\u6570\u3067\u3042\u308b\u3068\u306f, \u3042\u308b$w \\in \\mathbb{R}^r$\u3068$b \\in \\mathbb{R}$\u304c\u5b58\u5728\u3057\u3066, \n \\[\n f(x) = w^{\\mathrm{T}}x + b ~~~(x \\in \\mathbb{R}^r)\n \\]\n \u3068\u306a\u308b\u3053\u3068\u3092\u3044\u3046. \n\\end{defn}\n\n\\subsection{\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406\u306e\u6b74\u53f2}\n\u672c\u7bc0\u3067\u306f\u672c\u8ad6\u6587\u3067\u4e3b\u306b\u6271\u3046feedforward\u578b3\u5c64\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u5b9a\u7fa9\u3092\u4e0e\u3048, \u8a73\u3057\u304f\u6b74\u53f2\u3092\u632f\u308a\u8fd4\u308b.\n\u8a3c\u660e\u306f\u7ae0\u3092\u5909\u3048\u3066\u884c\u3046. \n\nfeedforward\u578b3\u5c64\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3092\u6b21\u306b\u3088\u308a\u5b9a\u7fa9\u3059\u308b.\n\\begin{defn}\n $r \\in \\mathbb{N}, W \\subset \\mathbb{R}^r, \\Theta \\subset \\mathbb{R}$\u3068\u95a2\u6570$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306b\u5bfe\u3057\u3066, \n \\[\n \\begin{aligned}\n &\\mathcal{N}_r(\\Psi,W,\\Theta) := \\mathrm{span} \\{\\mathbb{R}^r \\ni x \\mapsto \\Psi(w^{\\mathrm{T}} x + \\theta) \\in \\mathbb{R} \\mid w \\in W, \\theta \\in \\Theta \\} \\\\\n &\\Sigma^r(\\Psi) := \\mathcal{N}_r(\\Psi,\\mathbb{R}^r,\\mathbb{R}) = \\left\\{ \\sum_{j}^{q} \\beta_j \\Psi \\circ A_j \\mid  \\beta_j \\in \\mathbb{R}, A_j \\in \\mathbb{A}^r , q \\in \\mathbb{N}  \\right\\} \n \\end{aligned}\n \\]\n \u3068\u5b9a\u7fa9\u3059\u308b. \u3053\u308c\u3089\u3092$\\Psi$\u3092\u6d3b\u6027\u5316\u95a2\u6570\u3068\u3059\u308bfeedforward\u578b3\u5c64\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3068\u547c\u3076. \n\\end{defn}\n\u3053\u306e\u5b9a\u7fa9\u306b\u304a\u3044\u3066\u4e2d\u9593\u30e6\u30cb\u30c3\u30c8\u6570$q$\u306b\u5236\u9650\u3092\u8a2d\u3051\u306a\u3044\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b. \u3053\u306e\u4e8b\u60c5\u304b\u3089\u3053\u306e\u3088\u3046\u306a\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306fshallow wide network\u3068\u547c\u3070\u308c\u308b\u3053\u3068\u304c\u3042\u308b(\u306a\u304a, \u9006\u306b$q$\u306b\u5236\u9650\u3092\u8a2d\u3051, \u5c64\u306e\u6570\u306b\u5236\u9650\u3092\u8a2d\u3051\u306a\u3044\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af, deep narrow network\u306b\u3064\u3044\u3066\u3082\u8fd1\u5e74\u8003\u5bdf\u304c\u9032\u3081\u3089\u308c\u3066\u3044\u308b\\cite{DeepNarrowNetworks}). \n\n\u660e\u3089\u304b\u306b, $\\Psi \\in \\mathbb{M}(\\mathbb{R},\\mathbb{R})$, \u3064\u307e\u308a$\\Psi$\u304c$\\mathbb{R}$\u304b\u3089$\\mathbb{R}$\u3078\u306eBorel\u53ef\u6e2c\u95a2\u6570\u306a\u3089\u3070$\\Psi$\u3092\u6d3b\u6027\u5316\u95a2\u6570\u3068\u3059\u308bfeedforward\u578b3\u5c64\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306f$\\mathbb{M}(\\mathbb{R}^r,\\mathbb{R})$\u306b\u542b\u307e\u308c\u308b. \n\u307e\u305f, $\\Psi \\in C(\\mathbb{R},\\mathbb{R})$\u306a\u3089\u3070$\\Psi$\u3092\u6d3b\u6027\u5316\u95a2\u6570\u3068\u3059\u308bfeedforward\u578b3\u5c64\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306f$C(\\mathbb{R}^r,\\mathbb{R})$\u306b\u542b\u307e\u308c\u308b. \n\n\u6b21\u306b, feedforward\u578b3\u5c64\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406\u306e\u610f\u5473\u3092\u6b63\u78ba\u306b\u8ff0\u3079\u308b\u305f\u3081\u306b, \u8ddd\u96e2\u7a7a\u9593\u306b\u304a\u3051\u308b\u7a20\u5bc6\u6027\u306e\u5b9a\u7fa9\u3092\u3059\u308b. \n\\begin{defn}\n $(X,\\rho)$\u3092\u8ddd\u96e2\u7a7a\u9593\u3068\u3057, $S,T \\subset X$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $S$\u304c$T$\u306b\u304a\u3044\u3066$\\rho$-\u7a20\u5bc6\u3067\u3042\u308b\u3068\u306f, \u4efb\u610f\u306e$t \\in T$\u3068$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, \u3042\u308b$s \\in S$\u304c\u5b58\u5728\u3057\u3066, $\\rho(s,t) < \\varepsilon$\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u3044\u3046. \n\\end{defn}\n\u306a\u304a, \u3057\u3070\u3057\u3070\u3053\u306e\u6982\u5ff5\u3092\u6feb\u7528\u3057\u3066,  \u5fc5\u305a\u3057\u3082$X$\u306b\u542b\u307e\u308c\u306a\u3044\u96c6\u5408$S,T$\u306b\u3064\u3044\u3066$\\rho$-\u7a20\u5bc6\u3068\u3044\u3046\u8868\u73fe\u3092\u3059\u308b\u3053\u3068\u304c\u3042\u308b(\u305d\u306e\u5834\u5408\u306f$S,T$\u306e\u5143\u306b\u3064\u3044\u3066$\\rho$\u306e\u5024\u304c\u5b9a\u307e\u308b\u3053\u3068\u3092\u524d\u63d0\u3068\u3059\u308b). \n\\begin{defn}\n  $K \\subset \\mathbb{R}^r$\u3092\u7a7a\u3067\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u3068\u3059\u308b. $f,g \\in C(K)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\rho_K(f,g) := \\sup_{x \\in K} |f(x) - g(x)|\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u307e\u305f, $S \\subset C(\\mathbb{R}^r)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  S_K := \\{f|_K \\mid f \\in S\\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u3053\u3053\u3067$f|_K$\u306f$f$\u306e$K$\u3078\u306e\u5236\u9650\u3067\u3042\u308b. $(S_K,\\rho_K)$\u306f\u8ddd\u96e2\u7a7a\u9593\u3068\u306a\u308b. \n\\end{defn}\n\\begin{defn}\n  $C(\\mathbb{R}^r)$\u306e\u5143\u306e\u5217$(f_n)_{n=1}^{\\infty}$\u304c$f \\in C(\\mathbb{R}^r)$\u306b\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u3059\u308b\u3068\u306f, \u4efb\u610f\u306e\u7a7a\u3067\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}^r$\u306b\u5bfe\u3057\u3066, $\\rho_K(f_n,f) \\rightarrow 0 ~(n \\rightarrow \\infty)$\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u3044\u3046. \u307e\u305f, $S \\subset C(\\mathbb{R}^r)$\u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3068\u306f, \u4efb\u610f\u306e\u7a7a\u3067\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}^r$\u306b\u5bfe\u3057\u3066, $S_K$\u304c$C(K)$\u306b\u304a\u3044\u3066$\\rho_K$-\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3092\u3044\u3046. \n\\end{defn}\n\n1960\u5e74\u4ee3\u4ee5\u964d\u306e\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u5fdc\u7528\u306b\u304a\u3051\u308b\u6570\u591a\u304f\u306e\u6210\u529f\u304b\u3089, \u3069\u306e\u3088\u3046\u306a\u6761\u4ef6\u4e0b\u3067$\\mathcal{N}_r(\\Psi,W,\\Theta)$\u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u304b?\u3068\u3044\u3046\u554f\u3044\u304c\u7acb\u3066\u3089\u308c\u305f. Hornik\\cite{Hornik}\u306b\u3088\u308c\u3070, \u3053\u306e\u554f\u3044\u3092\u7814\u7a76\u8005\u304c\u7814\u7a76\u3057\u59cb\u3081\u305f\u306e\u306f1980\u5e74\u4ee3\u306b\u306a\u3063\u3066\u304b\u3089\u3067\u3042\u308b. \n\u6b74\u53f2\u7684\u306b\u306f\u307e\u305a$\\Sigma^r(\\Psi)$\u306b\u95a2\u3057\u3066\u8003\u5bdf\u304c\u9032\u3081\u3089\u308c\u305f. \n\u3064\u307e\u308a, \u6d3b\u6027\u5316\u95a2\u6570$\\Psi$\u304c\u3069\u306e\u3088\u3046\u306a\u3082\u306e\u306a\u3089\u3070\u4e07\u80fd\u8fd1\u4f3c\u80fd\u529b\u3092\u6301\u3064\u304b\u3068\u3044\u3046\u554f\u984c\u304c\u8003\u3048\u3089\u308c\u305f. \n\u3053\u306e\u554f\u984c\u306f1989\u5e74\u306b\u8907\u6570\u306e\u7814\u7a76\u8005\u306b\u3088\u308a\u72ec\u7acb\u306b\u7570\u306a\u308b\u624b\u6cd5\u3067\u89e3\u304b\u308c\u305f.\nCybenko\\cite{Cybenko}\u306f$\\Psi$\u304c\u9023\u7d9a\u306asigmoidal\u95a2\u6570\u3067\u3042\u308b\u3068\u304d\u306b\u4e07\u80fd\u8fd1\u4f3c\u80fd\u529b\u3092\u6301\u3064\u3053\u3068\u3092Hahn-Banach\u306e\u62e1\u5f35\u5b9a\u7406\u3084Riesz-Markov-\u89d2\u8c37\u306e\u8868\u73fe\u5b9a\u7406\u306a\u3069\u3092\u7528\u3044\u3066\u793a\u3057\u305f. \n\u3053\u3053\u3067, sigmoidal\u95a2\u6570\u306e\u5b9a\u7fa9\u306f\u6b21\u306e\u901a\u308a\u3067\u3042\u308b. \n\n\\begin{defn}[sigmoidal\u95a2\u6570] \\\\\n \u95a2\u6570 $\\sigma : \\mathbb{R} \\rightarrow \\mathbb{R}$ \u304c\u6b21\u306e\u6761\u4ef6\u3092\u307f\u305f\u3059\u3068\u304d, sigmoidal\u95a2\u6570\u3068\u547c\u3076:\n \\[\n \\sigma(t) \\rightarrow\n \\begin{cases}\n 1 & (t \\rightarrow +\\infty)\\\\\n 0 & (t \\rightarrow -\\infty)\n \\end{cases}\n \\]\n\\end{defn}\n\n\u4e00\u65b9, Hornik\\cite{Hornik}\u306f$\\Psi$\u304csquashing\u95a2\u6570(\u5358\u8abf\u975e\u6e1b\u5c11\u306asigmoidal\u95a2\u6570)\u306e\u5834\u5408\u3092Stone-Wierstrass\u306e\u5b9a\u7406\u3092\u4f7f\u3063\u3066\u8a3c\u660e\u3057\u305f.\n\u307e\u305f, \u8239\u6a4b\\cite{FUNAHASHI1989}\u306f$\\Psi$\u304c\u975e\u5b9a\u6570\u304b\u3064\u6709\u754c\u304b\u3064\u5358\u8abf\u975e\u6e1b\u5c11\u306a\u9023\u7d9a\u95a2\u6570\u306e\u5834\u5408\u3092\u5165\u6c5f\u30fb\u4e09\u5b85\\cite{IrieMiyake1988}\u304c\u793a\u3057\u305f\u4e8b\u5b9f\u3092\u7528\u3044\u3066\u8a3c\u660e\u3057\u305f. \n\u305d\u3057\u3066, 1993\u5e74\u306b\u306fLeshno et al. \\cite{Leshno} \u306b\u3088\u3063\u3066\u4ee5\u4e0b\u306e\u5b9a\u7406\u304c\u8a3c\u660e\u3055\u308c\u305f.  \n\n\\begin{defn*}\n  $\\Omega \\subset \\mathbb{R}^n$\u3092\u958b\u96c6\u5408\u3068\u3059\u308b. \u307e\u305f,  $\\mu$\u3092$(\\mathbb{R}^n,\\mathcal{B}_{\\mathbb{R}^n})$\u4e0a\u306eLebesgue\u6e2c\u5ea6\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \n  \\[\n  \\begin{aligned}\n  L_{\\mathrm{loc}}^{\\infty}(\\Omega) &:= \\{f \\mid \\mbox{\u4efb\u610f\u306e\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408}K \\subset \\Omega \\mbox{\u306b\u5bfe\u3057\u3066}f|_{K} \\in L^{\\infty}(K) \\} \\\\\n  \\mathcal{M}(\\Omega) &:= \\{f \\in L_{\\mathrm{loc}}^{\\infty}(\\Omega) \\mid \\mu(\\mathrm{cl}\\{f\\mbox{\u306e\u4e0d\u9023\u7d9a\u70b9}\\}) = 0\\}\n  \\end{aligned}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \n\\end{defn*}\n\\begin{thm*}[Leshno et al. 1993 \\cite{Leshno}]\n  $\\mu$\u3092$\\mathbb{R}$\u4e0a\u306eLebesgue\u6e2c\u5ea6\u3068\u3059\u308b. $\\Psi \\in \\mathcal{M}(\\mathbb{R})$\u306b\u5bfe\u3057\u3066, $\\Sigma^r(\\Psi)$\u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u305f\u3081\u306e\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306f, \u3044\u304b\u306a\u308b\u4e00\u5909\u6570\u591a\u9805\u5f0f\u95a2\u6570$P:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306b\u3064\u3044\u3066\u3082$\\Psi(x) = P(x) ~(\\mu \\mathrm{-a.e.}x)$\u3068\u306a\u3089\u306a\u3044\u3053\u3068\u3067\u3042\u308b. \n\\end{thm*}\n\u3055\u3089\u306b, \u3053\u306e\u5b9a\u7406\u304c\u767a\u8868\u3055\u308c\u305f\u306e\u3061, Hornik\\cite{Hornik1993}\u306b\u3088\u308a\u9023\u7d9a\u6027\u306b\u95a2\u3059\u308b\u4eee\u5b9a\u3092\u7de9\u548c\u3057\u305f\u6b21\u306e\u7d50\u679c\u304c\u5f97\u3089\u308c\u305f. \n\\begin{thm*}[Hornik 1993 \\cite{Hornik1993}]\n  $\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f\u5404\u9589\u533a\u9593\u4e0a\u3067\u6709\u754c\u304b\u3064Riemann\u7a4d\u5206\u53ef\u80fd\u3067\u3042\u308b\u3068\u3059\u308b\u3068,  $\\Sigma^r(\\Psi)$\u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3068, $\\Psi$\u304c\u591a\u9805\u5f0f\u95a2\u6570\u3068a.e.\u3067\u4e00\u81f4\u3057\u306a\u3044\u3053\u3068\u306f\u540c\u5024\u3067\u3042\u308b. \n\\end{thm*}\n\u9589\u533a\u9593\u4e0a\u3067\u6709\u754c\u306aRiemann\u53ef\u7a4d\u5206\u95a2\u6570\u306fLebesgue\u53ef\u6e2c\u304b\u3064\u305d\u306e\u4e0d\u9023\u7d9a\u70b9\u5168\u4f53\u306e\u96c6\u5408\u306e\u6e2c\u5ea6\u304c$0$\u306b\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b(\u732a\u72e9\\cite{igarizitukaiseki}\u5b9a\u7406$3$.$24$\u3092\u53c2\u7167\u3055\u308c\u305f\u3044). \nLeshno et al.\u306e\u7d50\u679c\u3067\u306f\u300c\u4e0d\u9023\u7d9a\u70b9\u5168\u4f53\u306e\u96c6\u5408\u306e\u9589\u5305\u300d\u306e\u6e2c\u5ea6\u304c$0$\u306b\u306a\u308b\u3053\u3068\u304c\u4eee\u5b9a\u3055\u308c\u3066\u3044\u305f\u306e\u3067, \u78ba\u304b\u306b\u9023\u7d9a\u6027\u306b\u95a2\u3059\u308b\u4eee\u5b9a\u304c\u7de9\u548c\u3055\u308c\u3066\u3044\u308b. \n\nLeshno et al.\u3084Hornik\u306b\u3088\u308a\u793a\u3055\u308c\u305f\u3053\u308c\u3089\u306e\u5b9a\u7406\u306e\u9069\u7528\u7bc4\u56f2\u306f\u5e83\u304f, \u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406\u3092\u3081\u3050\u308b\u4e00\u9023\u306e\u8b70\u8ad6\u306b\u306f\u4e00\u5fdc\u306e\u6c7a\u7740\u304c\u4ed8\u3051\u3089\u308c\u305f\u3068\u8003\u3048\u3066\u3088\u3044\u3060\u308d\u3046. \u305f\u3060\u3057, \u4ee5\u4e0a\u3067\u8ff0\u3079\u305f\u7d50\u679c\u306e\u3046\u3061, 1989\u5e74\u306eCybenko, Hornik\u304a\u3088\u30731993\u5e74\u306eLeshno et al., Hornik\u306e\u7d50\u679c\u306f\u9069\u7528\u3067\u304d\u308b\u7bc4\u56f2\u306b\u5171\u901a\u90e8\u5206\u306f\u3042\u308c\u3069\u5305\u542b\u95a2\u4fc2\u306f\u7121\u3044. \n\n\u4e07\u80fd\u8fd1\u4f3c\u80fd\u529b\u304c\u660e\u3089\u304b\u306b\u306a\u3063\u3066\u304b\u3089\u306f\u8fd1\u4f3c\u30ec\u30fc\u30c8\u306e\u554f\u984c(\u4e2d\u9593\u30e6\u30cb\u30c3\u30c8\u306e\u6570\u3068\u8fd1\u4f3c\u7cbe\u5ea6\u306e\u95a2\u4fc2)\u306b\u7126\u70b9\u304c\u79fb\u308a, Mhaskar\u306b\u3088\u308bSobolev\u7a7a\u9593\u306b\u304a\u3051\u308b\u8fd1\u4f3c\u30ec\u30fc\u30c8\u306e\u7814\u7a76\\cite{Mhaskar1995}\u3084Barron\\cite{Barron1993}\u306b\u59cb\u307e\u308bBarron space\u3068\u547c\u3070\u308c\u308b\u300c\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3067\u3088\u304f\u8fd1\u4f3c\u3067\u304d\u308b\u7a7a\u9593\u300d\u306e\u7814\u7a76\u306a\u3069\u8907\u6570\u306e\u30a2\u30d7\u30ed\u30fc\u30c1\u3067\u306e\u7d50\u679c\u304c\u51fa\u3066\u304a\u308a, \u73fe\u5728\u3067\u3082\u7814\u7a76\u304c\u7d9a\u3044\u3066\u3044\u308b(Weinan et al.\\cite{Weinan2019}\u3084Siegel, Xu\\cite{Siegel2020}\u3092\u53c2\u7167\u3055\u308c\u305f\u3044). \n\n\u672c\u8ad6\u6587\u3067\u306f\u307e\u305a\u7b2c3\u7ae0\u3067\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406\u3068\u95a2\u308f\u308a\u304c\u6df1\u3044\u30ea\u30c3\u30b8\u95a2\u6570\u306e\u6027\u8cea\u306b\u3064\u3044\u3066\u3044\u304f\u3064\u304b\u8ff0\u3079\u308b. \n\u305d\u3057\u3066\u7b2c4\u7ae0\u306e\u306f\u3058\u3081\u306b1993\u5e74\u306eLeshno et al., Hornik\u306e\u7d50\u679c\u3092\u8a3c\u660e\u3059\u308b. \n\u6b21\u3044\u3067, Cybenko\u306e\u7d50\u679c\u3092\u4e00\u822c\u5316\u3057\u305fChui\u3068Li\u306e\u7d50\u679c\\cite{ChuiAndLi}(\u4e0b\u8868\u53c2\u7167)\u3092\u89e3\u8aac\u3059\u308b. \n\u3055\u3089\u306bChui\u3068Li\u306e\u7d50\u679c\u3092\u30ce\u30eb\u30e0\u7a7a\u9593\u3078\u4e00\u822c\u5316\u3057\u305f, Sun\u3068Cheney\u306b\u3088\u308b\u7d50\u679c\\cite{SunAndCheney}\u3092\u7d39\u4ecb\u3057, \u305d\u306e\u5f8c, 1989\u5e74\u306eHornik\u306e\u7d50\u679c\u306e\u8a3c\u660e\u3092\u4e0e\u3048\u308b. \n\n\u6b21\u306b\u7ae0\u3092\u5909\u3048, 5\u7ae0\u3067\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3051\u308b\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406\u306e\u5fdc\u7528\u3068\u3057\u3066\u4ee5\u4e0a\u306e\u7d50\u679c\u3092\u591a\u5c64\u30fb\u591a\u51fa\u529b\u306b\u3057\u305f\u5834\u5408\u3084$L^p$\u7a7a\u9593\u306b\u304a\u3051\u308b\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406, \u95a2\u6570\u306e\u88dc\u9593\u306b\u3064\u3044\u3066\u8ff0\u3079\u308b. \n\u3055\u3089\u306b, 6\u7ae0\u3067\u306ffeedforward\u578b\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u304b\u3089\u4e00\u65e6\u96e2\u308cRBF(Radial-Basis-Function)\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406\u3092\u8a3c\u660e\u3059\u308b. \n\u305d\u306e\u5f8c, 7\u7ae0\u3067\u306f\u8a71\u984c\u3092\u8fd1\u4f3c\u30ec\u30fc\u30c8\u306e\u554f\u984c\u306b\u79fb\u3057,  feedforward\u578b3\u5c64\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u8fd1\u4f3c\u8aa4\u5dee\u89e3\u6790\u306b\u3064\u3044\u3066\u89e3\u8aac\u3059\u308b. \n  \\begin{table}[htb]\n    \\caption{$\\mathcal{N}_r(\\Psi,W,B)$\u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u4e07\u80fd\u8fd1\u4f3c\u80fd\u529b\u3092\u6301\u3064\u6761\u4ef6}\n    \\small\n    \\begin{tabular}{|c|c|c|c|c|c|c|} \\hline\n     {\\scriptsize \u897f\u66a6}& \u8a3c\u660e\u8005 & $\\Psi$ & $\\Psi$\u306e\u9023\u7d9a\u6027 & $W$ & $B$ & \\begin{tabular}{c} {\\tiny \u672c\u8ad6\u6587} \\\\ {\\tiny \u5b9a\u7406\u756a\u53f7} \\end{tabular} \\\\ \\hline \n     & Cybenko & sigmoidal & \u9023\u7d9a & $\\mathbb{R}^r$ & $\\mathbb{R}$ & {\\scriptsize 4.2.1} \\\\ \\cline{2-7}\n    {\\scriptsize 1989} & {\\footnotesize Funahashi} & \\begin{tabular}{c} \u6709\u754c \\\\ \u5358\u8abf\u975e\u6e1b\u5c11 \\end{tabular} & \u9023\u7d9a & $\\mathbb{R}^r$ & $\\mathbb{R}$ & {\\scriptsize 4.1.11} \\\\\\cline{2-7}\n     & Hornik & \\begin{tabular}{c} sigmoidal \\\\ \u5358\u8abf\u975e\u6e1b\u5c11 \\end{tabular} & \u4eee\u5b9a\u306a\u3057 & $\\mathbb{R}^r$ & $\\mathbb{R}$ & {\\scriptsize 4.4.6} \\\\ \\hline\n    {\\scriptsize 1992} & Chui, Li & sigmoidal & \u9023\u7d9a & $\\mathbb{Z}^r$ & $\\mathbb{Z}$ & {\\scriptsize 4.2.1} \\\\ \\hline\n    {\\scriptsize 1993} & {\\scriptsize Leshno et al.} & {\\footnotesize \u975e\u591a\u9805\u5f0f, $L_{\\mathrm{loc}}^{\\infty}$} & { \\footnotesize $\\mu\\left(\\mathrm{cl}\\{\\mbox{\u4e0d\u9023\u7d9a\u70b9}\\}\\right) = 0$ } & $\\mathbb{R}^r$ & $\\mathbb{R}$ & {\\scriptsize 4.1.11} \\\\ \\cline{2-7}\n     & Hornik & \\begin{tabular}{c} {\\footnotesize $B$\u4e0a\u975e\u591a\u9805\u5f0f}\\\\ \u5c40\u6240\u6709\u754c \\end{tabular} & {\\footnotesize $\\mu\\left(\\{\\mbox{\u4e0d\u9023\u7d9a\u70b9}\\}\\right)=0$ } & {\\footnotesize $0 \\in W^{\\circ}$} & {\\footnotesize \u958b\u533a\u9593} & {\\scriptsize 4.1.12} \\\\ \\hline\n    \\end{tabular}\n  \\end{table}\n\n\\section{\u30ea\u30c3\u30b8\u95a2\u6570\u3068\u305d\u306e\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406\u3078\u306e\u5fdc\u7528}\n\u672c\u7ae0\u3067\u306f\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406\u3068\u95a2\u308f\u308a\u304c\u6df1\u3044\u30ea\u30c3\u30b8\u95a2\u6570\u306e\u6027\u8cea\u306b\u3064\u3044\u3066\u8ff0\u3079, \u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406\u3092\u8a3c\u660e\u3059\u308b\u305f\u3081\u306b\u306f$1$\u6b21\u5143\u306e\u5834\u5408\u306e\u307f\u8003\u3048\u308c\u3070\u5341\u5206\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059. \n\\subsection{\u30ea\u30c3\u30b8\u95a2\u6570\u306e\u6027\u8cea}\n\n\\begin{defn}[\u6709\u754c\u7dda\u5f62\u5199\u50cf, \u5171\u5f79\u7a7a\u9593] \\\\\n  $X,Y$\u3092\u30ce\u30eb\u30e0\u7a7a\u9593\u3068\u3059\u308b. $f:X \\rightarrow Y$\u304c\u6709\u754c\u7dda\u5f62\u5199\u50cf\u3067\u3042\u308b\u3068\u306f, $f$\u304c\u7dda\u5f62\u5199\u50cf\u3067\u3042\u3063\u3066, \u304b\u3064$M \\geq 0$\u304c\u5b58\u5728\u3057\u3066, \u4efb\u610f\u306e$x \\in X$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\lVert f(x) \\rVert_Y \\leq M \\lVert x \\rVert_X\n  \\]\n  \u3068\u306a\u308b\u3053\u3068\u3092\u3044\u3046. \u6709\u754c\u7dda\u5f62\u5199\u50cf\u306f\u30ce\u30eb\u30e0\u306b\u95a2\u3057\u3066\u9023\u7d9a\u3067\u3042\u308b. \u307e\u305f, $X$\u306e\u5171\u5f79\u7a7a\u9593\u3092 \n  \\[\n  X^* := \\{f:X \\rightarrow \\mathbb{R} \\mid f\\mbox{\u306f\u6709\u754c\u7dda\u5f62\u5199\u50cf}\\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u305d\u3057\u3066, $f \\in X^*$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\lVert f \\rVert = \\inf \\{M \\geq 0 \\mid \\forall x \\in X, | f(x) | \\leq M \\lVert x \\rVert_X \\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. $\\lVert \\cdot \\rVert$\u306f$X^*$\u4e0a\u306e\u30ce\u30eb\u30e0\u3067\u3042\u308a, \u4f5c\u7528\u7d20\u30ce\u30eb\u30e0\u3068\u547c\u3070\u308c\u308b. \n\\end{defn}\n\\begin{defn}[\u30ea\u30c3\u30b8\u95a2\u6570] \\\\\n  $X$\u3092\u30ce\u30eb\u30e0\u7a7a\u9593\u3068\u3059\u308b. \u95a2\u6570$f:X \\rightarrow \\mathbb{R}$\u304c\u30ea\u30c3\u30b8\u95a2\u6570\u3067\u3042\u308b\u3068\u306f, $\\varphi \\in X^*$\u3068$g:\\mathbb{R} \\rightarrow \\mathbb{R}$\u304c\u5b58\u5728\u3057\u3066, $f = g \\circ \\varphi$\u3068\u306a\u308b\u3053\u3068\u3092\u3044\u3046. \u7279\u306b$X$\u304cHilbert\u7a7a\u9593\u3067\u3042\u308b\u3068\u304d\u306fRiesz\u306e\u8868\u73fe\u5b9a\u7406\u3088\u308a, $v \\in X$\u3068$g:\\mathbb{R} \\rightarrow \\mathbb{R}$\u304c\u5b58\u5728\u3057\u3066, $f(x) = g(\\ip<{x,v}>)$\u3068\u306a\u308b\u3053\u3068\u3092\u3044\u3046. \n\\end{defn}\n\\begin{defn}\n  \u30ce\u30eb\u30e0\u7a7a\u9593$X$\u3068$\\mathcal{F} \\subset X^*$\u306b\u5bfe\u3057\u3066, \n  \\[\n  C(\\mathbb{R}) \\circ \\mathcal{F} = \\{g \\circ f \\mid g \\in C(\\mathbb{R}), f \\in \\mathcal{F}\\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. $C(\\mathbb{R}) \\circ \\mathcal{F} \\subset C(X)$\u3067\u3042\u308b. \n\\end{defn}\n\\begin{lem}\\label{VandermondeCor}\n  $r,s \\in \\mathbb{N} \\cup \\{0\\}$\u3068\u76f8\u7570\u306a\u308b$\\beta_j \\in \\mathbb{R} \\setminus \\{0\\} ~~(j=0,\\ldots,r+s)$\u306b\u5bfe\u3057\u3066, \u3042\u308b$c_j \\in \\mathbb{R} ~~(j=0,\\ldots,r+s)$\u304c\u5b58\u5728\u3057\u3066, \u4efb\u610f\u306e$(x_1,x_2) \\in \\mathbb{R}^2$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\sum_{j=0}^{r+s} c_j (x_1 + \\beta_j x_2)^{r+s} = x_1^r x_2^s\n  \\]\n  \u3068\u306a\u308b. \n\\end{lem}\n\\begin{proof}\n  \u3082\u3057, \u305d\u306e\u3088\u3046\u306a$c_j \\in \\mathbb{R}$\u304c\u5b58\u5728\u3059\u308b\u306a\u3089, \u4efb\u610f\u306e$(x_1,x_2) \\in \\mathbb{R}^2$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  \\sum_{j=0}^{r+s} c_j (x_1 + \\beta_j x_2)^{r+s}\n  &= \\sum_{k=0}^{r+s} \\binom{r+s}{k} \\sum_{j=0}^{r+s}  c_j \\beta_j^k x_1^{r+s-k} x_2^k \\\\\n  &= \\sum_{k=0}^{r+s} \\delta_{k,s} x_1^{r+s-k} x_2^{k}\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b\u306e\u3067, $c_j$\u306f\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\n  \\[\n  \\binom{r+s}{k} \\sum_{j=0}^{r+s} c_j \\beta_j^k = \\delta_{k,s} ~~(k=0,\\ldots,r+s)\n  \\]\n  \u3092\u307f\u305f\u3059. \u3068\u3053\u308d\u304c, \u3053\u306e\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306fVandermonde\u884c\u5217\u306e\u6b63\u5247\u6027\u304b\u3089\u4e00\u610f\u7684\u306b\u89e3\u3051\u308b($\\beta_j$\u306f\u76f8\u7570\u306a\u308a, \u304b\u3064\u3059\u3079\u3066$0$\u3067\u306a\u3044\u3053\u3068\u306b\u6ce8\u610f). \u305d\u3057\u3066, \u3053\u306e\u89e3$c_j$\u306f\u6240\u671b\u306e\u6027\u8cea\u3092\u6301\u3064. \n\\end{proof}\n\\begin{thm}\\label{densityRidgeFunc}\n  $X$\u3092\u30ce\u30eb\u30e0\u7a7a\u9593\u3068\u3059\u308b\u3068\u304d,  $X$\u4e0a\u306e\u30ea\u30c3\u30b8\u95a2\u6570\u5168\u4f53\u306e\u96c6\u5408\u306e\u7dda\u5f62\u5305\u306f$C(X)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{thm}\n\\begin{proof}\n  \u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset X$\u306b\u5bfe\u3057\u3066, \u90e8\u5206\u7a7a\u9593$P \\subset C(K)$\u3092\u6b21\u3067\u5b9a\u3081\u308b. \n  \\[\n  P = \\mathrm{span}\\{K \\ni x \\mapsto (\\varphi(x))^n \\in \\mathbb{R} \\mid n \\in \\mathbb{N} \\cup \\{0\\} , ~\\varphi \\in X^*\\}\n  \\]\n  \u3059\u308b\u3068, $P$\u306f\u901a\u5e38\u306e\u548c\u3068\u30b9\u30ab\u30e9\u30fc\u500d, \u304a\u3088\u3073\u7a4d\u306b\u95a2\u3057\u3066\u4ee3\u6570\u3092\u306a\u3059.\n  \u305d\u306e\u3053\u3068\u3092\u793a\u3059\u306b\u306f, \u4efb\u610f\u306e$\\varphi,\\theta \\in X^*$\u3068$r,s \\in \\mathbb{N} \\cup \\{0\\}$\u306b\u5bfe\u3057\u3066, $x \\mapsto \\varphi(x)^r \\theta(x)^s$\u304c$P$\u306b\u5c5e\u3059\u308b\u3053\u3068\u3092\u8a00\u3048\u3070\u3088\u3044\u304c, \u88dc\u984c\\ref{VandermondeCor}\u3088\u308a, $\\beta_j,c_j \\in \\mathbb{R} ~~(j=0,\\ldots,r+s)$\u304c\u5b58\u5728\u3057\u3066, \u4efb\u610f\u306e$x \\in K$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\varphi(x)^r \\theta(x)^s = \\sum_{j=0}^{r+s} c_j (\\varphi(x) + \\beta_j \\theta(x))^{r+s}\n  \\]\n  \u3068\u306a\u308b\u306e\u3067$x \\mapsto \\varphi(x)^r \\theta(x)^s$\u306f$P$\u306b\u5c5e\u3059\u308b. \u305d\u3057\u3066, $x,y \\in K$\u304c$x \\neq y$\u3092\u307f\u305f\u3059\u306a\u3089\u3070, Hahn-Banach\u306e\u62e1\u5f35\u5b9a\u7406\u3088\u308a$\\varphi(x) \\neq \\varphi(y)$\u306a\u308b$\\varphi \\in X^*$\u304c\u53d6\u308c\u308b. \u3055\u3089\u306b, $n$\u3068\u3057\u3066$0$\u3092\u3068\u308b\u3053\u3068\u3067$1 \\in P$\u304c\u308f\u304b\u308b. \u3088\u3063\u3066, Stone-Weierstrass\u306e\u5b9a\u7406\u3088\u308a,  $P$\u306f$C(K)$\u306b\u304a\u3044\u3066\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \u3053\u308c\u3088\u308a$X$\u4e0a\u306e\u30ea\u30c3\u30b8\u95a2\u6570\u5168\u4f53\u306e\u96c6\u5408\u306e\u7dda\u5f62\u5305\u306f$C(X)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{proof}\n\\begin{lem}\n  $V = \\{\\mathbb{R}^r \\ni x \\mapsto \\sum_{i=1}^l f_i(a_i^{\\mathrm{T}}x) \\in \\mathbb{R} \\mid l \\in \\mathbb{N} , a_i \\in \\mathbb{R}^r , f_i \\in C(\\mathbb{R}) \\}$\u3068\u304a\u304f\u3068, $V$\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{lem}\n\\begin{proof}\n  \u5b9a\u7406\\ref{densityRidgeFunc}\u306b\u3088\u308b(\u9023\u7d9a\u95a2\u6570\u3092\u5b9a\u6570\u500d\u3057\u3066\u3082\u9023\u7d9a\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f). \n\\end{proof}\n\u6b21\u306e\u547d\u984c\u306b\u3088\u308a, \u9023\u7d9a\u95a2\u6570\u306e\u7a7a\u9593\u3067\u306e\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af$\\Sigma^r(\\Psi)$\u306e\u7a20\u5bc6\u6027\u3092\u8003\u3048\u308b\u4e0a\u3067\u306f$1$\u6b21\u5143\u306e\u5834\u5408\u3092\u8003\u3048\u308c\u3070\u5341\u5206\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \n\\begin{prop}\\label{DensityDim1Suff}\n  \u4efb\u610f\u306e$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306b\u5bfe\u3057\u3066, $\\Sigma^1(\\Psi)$\u304c$C(\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u306a\u3089\u3070, \u4efb\u610f\u306e$r$\u306b\u3064\u3044\u3066$\\Sigma^r(\\Psi)$\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{prop}\n\\begin{proof}\n  $g \\in C(\\mathbb{R}^r)$\u3068\u3057, \u4efb\u610f\u306b\u7a7a\u3067\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}^r$\u3092\u3068\u308b. \u4efb\u610f\u306e$\\varepsilon>0$\u306b\u5bfe\u3057\u3066, \u3059\u3050\u4e0a\u306e\u88dc\u984c\u306b\u3088\u308a$k \\in \\mathbb{N}$\u3068$a_i \\in \\mathbb{R}^r, f_i \\in C(\\mathbb{R}) ~(i=1,\\ldots,k)$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\sup_{x \\in K} |g(x) - \\sum_{i=1}^{k} f_i(a_i^{\\mathrm{T}}x)| < \\varepsilon\/2\n  \\]\n  \u3068\u306a\u308b. $K$\u306f\u30b3\u30f3\u30d1\u30af\u30c8\u306a\u306e\u3067, \u5404$i$\u306b\u3064\u3044\u3066$\\{a_i^{\\mathrm{T}}x \\mid x \\in K\\} \\subset [\\alpha_i,\\beta_i]$\u3092\u307f\u305f\u3059$\\alpha_i,\\beta_i \\in \\mathbb{R}$\u304c\u53d6\u308c\u308b. \u305d\u3053\u3067, $\\Sigma^1(\\Psi)$\u306e\u7a20\u5bc6\u6027\u304b\u3089, $A_{i,j} \\in \\mathbb{A}^1$\u3068$c_{i,j} \\in \\mathbb{R}$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\sup_{y \\in [\\alpha_i,\\beta_i]} |f_i(y) - \\sum_{j=1}^{m_i} c_{i,j}\\Psi(A_{i,j}(y))| < \\varepsilon\/2k\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, \u4efb\u610f\u306e$x \\in K$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  &|g(x) - \\sum_{i=1}^k \\sum_{j=1}^{m_i} c_{i,j}\\Psi(A_{i,j}(a_i^{\\mathrm{T}}x))| \\\\\n  &\\leq |g(x) - \\sum_{i=1}^{k} f(a_i^{\\mathrm{T}}x)| + \\sum_{i=1}^k |f_i(a_i^{\\mathrm{T}}x) - \\sum_{j=1}^{m_i} c_{i,j}\\Psi(A_{i,j}(a_i^{\\mathrm{T}}x))| \\\\\n  &< \\varepsilon\/2 + \\varepsilon\/2 = \\varepsilon\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3057\u304b\u3082$x \\mapsto A_{i,j}(a_i^{\\mathrm{T}}x)$\u306f$\\mathbb{A}^r$\u306b\u5c5e\u3059\u308b. \n\\end{proof}\n\u3055\u3089\u306b\u30ea\u30c3\u30b8\u95a2\u6570\u306b\u3064\u3044\u3066\u6b21\u306e\u5b9a\u7406\u304c\u6210\u308a\u7acb\u3064. \n\\begin{thm}[Lin and Pinkus  \\cite{LinAndPinkus}]\\label{LinAndPinkus} \\\\\n  $\\mathcal{A} \\subset \\mathbb{R}^r$\u306b\u5bfe\u3057\u3066, \n  \\[\n  R(\\mathcal{A}) = \\{ \\mathbb{R}^r \\ni x \\mapsto f(w^{\\mathrm{T}} x) \\in \\mathbb{R} \\mid f \\in C(\\mathbb{R}),w \\in A\\}\n  \\]\n  \u3068\u304a\u304f\u3068, $\\mathrm{span}R(\\mathcal{A})$\u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3068, $\\mathcal{A}$\u4e0a\u3067$0$\u3092\u53d6\u308b\u975e\u81ea\u660e\u306a$r$\u5909\u6570\u6589\u6b21\u591a\u9805\u5f0f\u304c\u5b58\u5728\u3057\u306a\u3044\u3053\u3068\u306f\u540c\u5024\u3067\u3042\u308b. \n\\end{thm}\n\\begin{proof}\n  \u6b21\u7bc0\u53c2\u7167. \n\\end{proof}\n\u3053\u306e\u5b9a\u7406\u306b\u3088\u308a\u6b21\u306e\u3053\u3068\u304c\u308f\u304b\u308b. \n\\begin{prop}\\label{MultiVariateDensity}\n  $\\Psi: \\mathbb{R} \\rightarrow \\mathbb{R}$\u3068$W,B \\subset \\mathbb{R}$\u306b\u5bfe\u3057\u3066,  \n  \\[\n  \\mathcal{N}_1(\\Psi,W,B) = \\mathrm{span}\\{x \\mapsto \\Psi(wx+b) \\mid w \\in W, b \\in B\\}\n  \\]\n  \u306f$C(\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $\\mathcal{A} \\subset \\mathbb{R}^r$\u4e0a\u3067$0$\u3092\u53d6\u308b\u975e\u81ea\u660e\u306a$r$\u5909\u6570\u6589\u6b21\u591a\u9805\u5f0f\u304c\u5b58\u5728\u3057\u306a\u3044\u306a\u3089\u3070, \n  \\[\n  \\mathcal{N}_r(\\Psi,W\\mathcal{A},B) = \\mathrm{span}\\{\\Psi(aw^{\\mathrm{T}}x+b) \\mid a\\in \\mathcal{A}, w \\in W, b \\in B\\}\n  \\]\n  \u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u7a20\u5bc6\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{prop}\n\\begin{proof}\n  \u547d\u984c\\ref{DensityDim1Suff}\u3068\u540c\u69d8\u306b\u793a\u305b\u308b. \u8a73\u3057\u304f\u306f\u6b21\u306e\u901a\u308a\u3067\u3042\u308b. $g \\in C(\\mathbb{R}^r)$\u3068\u3057, \u4efb\u610f\u306b\u7a7a\u3067\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}^r$\u3092\u3068\u308b. \u4efb\u610f\u306e$\\varepsilon>0$\u306b\u5bfe\u3057\u3066, \u3059\u3050\u4e0a\u306e\u5b9a\u7406\u306b\u3088\u308a$k \\in \\mathbb{N}$\u3068$a_i \\in A, f_i \\in C(\\mathbb{R}) ~(i=1,\\ldots,k)$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\sup_{x \\in K} |g(x) - \\sum_{i=1}^{k} f_i(a_i^{\\mathrm{T}}x)| < \\varepsilon\/2\n  \\]\n  \u3068\u306a\u308b. $K$\u306f\u30b3\u30f3\u30d1\u30af\u30c8\u306a\u306e\u3067, \u5404$i$\u306b\u3064\u3044\u3066$\\{a_i^{\\mathrm{T}}x \\mid x \\in K\\} \\subset [\\alpha_i,\\beta_i]$\u3092\u307f\u305f\u3059$\\alpha_i,\\beta_i \\in \\mathbb{R}$\u304c\u53d6\u308c\u308b. \u305d\u3053\u3067, $\\mathcal{N}_1(\\Psi,W,B)$\u306e\u7a20\u5bc6\u6027\u304b\u3089, $m_i \\in \\mathbb{N}, w_{i,j} \\in W, b_{i,j} \\in B$\u3068$c_{i,j} \\in \\mathbb{R}$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\sup_{y \\in [\\alpha_i,\\beta_i]} |f_i(y) - \\sum_{j=1}^{m_i} c_{i,j}\\Psi(w_{i,j}y + b_{i,j})| < \\varepsilon\/2k\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, \u4efb\u610f\u306e$x \\in K$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  &|g(x) - \\sum_{i=1}^k \\sum_{j=1}^{m_i} c_{i,j}\\Psi(w_{i,j}(a_i^{\\mathrm{T}}x)+b_{i,j})| \\\\\n  &\\leq |g(x) - \\sum_{i=1}^{k} f(a_i^{\\mathrm{T}}x)| + \\sum_{i=1}^k |f_i(a_i^{\\mathrm{T}}x) - \\sum_{j=1}^{m_i} c_{i,j}\\Psi(w_{i,j}(a_i^{\\mathrm{T}}x) + b_{i,j})| \\\\\n  &< \\varepsilon\/2 + \\varepsilon\/2 = \\varepsilon\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \n\\end{proof}\n\n\\subsection{Lin\u3068Pinkus\u306e\u7d50\u679c\u306e\u8a3c\u660e}\n\u672c\u7bc0\u3067\u306f\u524d\u7bc0\u306b\u8ff0\u3079\u305fLin\u3068Pinkus\u306e\u7d50\u679c(\u5b9a\u7406\\ref{LinAndPinkus})\u306e\u8a3c\u660e\u3092\u8ff0\u3079\u308b. \n\\begin{defn}\n  $\\alpha = (\\alpha_1,\\ldots,\\alpha_n) \\in (\\mathbb{Z}_{\\geq 0})^n$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  |\\alpha| := \\sum_{j=1}^n \\alpha_j ,~~ \\alpha ! := \\alpha_1 ! \\cdots \\alpha_n !\n  \\end{aligned}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. $x=(x_1,\\ldots,x_n) \\in \\mathbb{R}^n$\u306b\u5bfe\u3057\u3066, \n  \\[\n  x^{\\alpha} := x_1^{\\alpha_1} \\cdots x_n^{\\alpha_n}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u504f\u5fae\u5206\u4f5c\u7528\u7d20$\\partial :=(\\partial_1,\\ldots,\\partial_n) =  (\\partial\/\\partial x_1, \\ldots, \\partial\/\\partial x_n)$\u306b\u5bfe\u3059\u308b$\\partial^{\\alpha}$\u3082\u540c\u69d8\u306b\u5b9a\u3081\u308b. \n\\end{defn}\n\\begin{defn}\n  $n,k \\in \\mathbb{N}$\u306b\u5bfe\u3057\u3066, $n$\u5909\u6570\u306e$k$\u6b21\u6589\u6b21\u591a\u9805\u5f0f\u95a2\u6570\u306e\u96c6\u5408\u3092\n  \\[\n  \\mathcal{H}_k(\\mathbb{R}^n) := \\left\\{ \\mathbb{R}^n \\ni x \\mapsto \\sum_{|\\alpha| = k} c_{\\alpha} x^{\\alpha} \\in \\mathbb{R} \\mid c_{\\alpha} \\in \\mathbb{R} \\right\\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u305f\u3060\u3057, \u548c\u306f$|\\alpha| = k$\u306a\u308b$\\alpha \\in (\\mathbb{Z}_{\\geq 0})^n$\u5168\u4f53\u306b\u308f\u305f\u308b\u3082\u306e\u3068\u3059\u308b. \u307e\u305f, \n  \\[\n  \\begin{aligned}\n  \\mathcal{P}_k(\\mathbb{R}^n) = \\bigcup_{s=0}^{k} \\mathcal{H}_s(\\mathbb{R}^n) ,~~ \\mathcal{P}(\\mathbb{R}^n) = \\bigcup_{s=0}^{\\infty} \\mathcal{H}_s(\\mathbb{R}^n)\n  \\end{aligned}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \n\\end{defn}\n\\begin{prop}\n  $\\mathcal{H}_k(\\mathbb{R}^n)$\u306f\u901a\u5e38\u306e\u548c\u3068\u30b9\u30ab\u30e9\u30fc\u500d\u306b\u95a2\u3057\u3066$\\mathbb{R}$\u4e0a\u306e\u6709\u9650\u6b21\u5143\u7dda\u5f62\u7a7a\u9593\u3092\u306a\u3059. \u307e\u305f, $\\mathcal{H}_k(\\mathbb{R}^n)$\u306e\u5143$p(x) = \\sum c_{\\alpha} x^{\\alpha},q(x) = \\sum c_{\\alpha}' x^{\\alpha} $\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\ip<{p,q}> := p(\\partial)q = \\sum_{|\\alpha| = k} \\alpha! c_{\\alpha} c_{\\alpha}'\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b\u3068, \u3053\u308c\u306f$\\mathcal{H}_k(\\mathbb{R}^n)$\u4e0a\u306e\u5185\u7a4d\u3067\u3042\u308a, \u3053\u306e\u5185\u7a4d\u306b\u95a2\u3057\u3066$\\mathcal{H}_k(\\mathbb{R}^n)$\u306fHilbert\u7a7a\u9593\u3092\u306a\u3059. \n\\end{prop}\n\\begin{proof}\n  $\\mathcal{H}_k(\\mathbb{R}^n)$\u304c$\\mathbb{R}$\u4e0a\u306e\u7dda\u5f62\u7a7a\u9593\u3067\u3042\u308b\u3053\u3068, $\\ip<{\\cdot,\\cdot}>$\u304c\u5185\u7a4d\u306e\u516c\u7406\u3092\u307f\u305f\u3059\u3053\u3068\u306f\u660e\u3089\u304b\u3067\u3042\u308b. \u6709\u9650\u6b21\u5143\u3067\u3042\u308b\u3053\u3068\u306f\u6709\u9650\u96c6\u5408$\\{ x^{\\alpha} \\mid |\\alpha| = k\\}$\u304c$\\mathcal{H}_k(\\mathbb{R}^n)$\u3092\u751f\u6210\u3059\u308b\u3053\u3068\u304b\u3089\u308f\u304b\u308b. \u5b8c\u5099\u6027\u306b\u3064\u3044\u3066\u306f, \u30ce\u30eb\u30e0\u7a7a\u9593\u306e\u6709\u9650\u6b21\u5143\u90e8\u5206\u7a7a\u9593\u304c\u5b8c\u5099\u3067\u3042\u308b\u3053\u3068\u306b\u3088\u308b(\\cite{FujitaKurodaIto}\u306e{\\S}1.5 (a)\u3092\u53c2\u7167\u3055\u308c\u305f\u3044). \n\\end{proof}\n\\begin{defn}\n  \u81ea\u7136\u6570$m,n \\geq 1$\u306b\u5bfe\u3057\u3066, $m \\times n$\u306e\u5b9f\u884c\u5217\u5168\u4f53\u306e\u96c6\u5408\u3092$\\mathrm{Mat}(m,n;\\mathbb{R})$\u3067\u8868\u3059. $A \\in \\mathrm{Mat}(m,n;\\mathbb{R})$\u306b\u5bfe\u3057\u3066, \n  \\[\n  L(A) := \\{ (y^{\\mathrm{T}}A)^{\\mathrm{T}} \\mid y \\in \\mathbb{R}^m \\} \\subset \\mathbb{R}^n\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. $\\Omega \\subset \\mathrm{Mat}(m,n;\\mathbb{R})$\u306b\u5bfe\u3057\u3066, \n  \\[\n  L(\\Omega) := \\bigcup_{A \\in \\Omega} L(A)\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u307e\u305f, \u30ea\u30c3\u30b8\u95a2\u6570\u306e\u96c6\u5408$R(\\Omega)$\u3092\n  \\[\n  R(\\Omega) = \\{\\mathbb{R}^n \\ni x \\mapsto g(Ax) \\in \\mathbb{R} \\mid A \\in \\Omega, g \\in C(\\mathbb{R}^m)\\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \n\\end{defn}\n\\begin{thm}[\u5c04\u5f71\u5b9a\u7406] \\\\\n  $X$\u3092Hilbert\u7a7a\u9593\u3068\u3057, $L \\subset X$\u3092\u9589\u90e8\u5206\u7a7a\u9593\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$x \\in X$\u306f\n  \\[\n  x = y + z ~~~(y \\in L,~ z \\in L^{\\perp})\n  \\]\n  \u306e\u5f62\u306b\u4e00\u610f\u7684\u306b\u5206\u89e3\u53ef\u80fd\u3067\u3042\u308b. \u305f\u3060\u3057, \n  \\[\n  L^{\\perp} := \\{x \\in X \\mid \\ip<{x,y}> = 0 ~(\\forall y \\in L)\\}\n  \\]\n  \u3067\u3042\u308b. \n\\end{thm}\n\\begin{proof}\n  \\cite{FujitaKurodaIto}\u306e\u5b9a\u7406$3$.$2$\u3092\u53c2\u7167\u3055\u308c\u305f\u3044. \n\\end{proof}\n\\begin{lem}\\label{HomPolyReprensent}\n  $k \\geq 0$\u3068\u3057$\\Omega \\subset \\mathrm{Mat}(n,m;\\mathbb{R})$\u3068\u3059\u308b. \n  \u307e\u305f, $L(\\Omega)$\u4e0a\u3067\u6052\u7b49\u7684\u306b$0$\u3092\u53d6\u308b$\\mathcal{H}_k(\\mathbb{R}^n)$\u306e\u5143\u306f$0$\u306b\u9650\u308b\u3068\u3059\u308b. \n  \u3053\u306e\u3068\u304d, \n  \\[\n  \\mathcal{H}_k(\\mathbb{R}^n) = \\mathrm{span}\\{ \\mathbb{R}^n \\ni x \\mapsto (d^{\\mathrm{T}}x)^k \\in \\mathbb{R} \\mid d \\in L(\\Omega)\\} =: L\n  \\]\n  \u3068\u306a\u308b. \n\\end{lem}\n\\begin{proof}\n  $\\mathcal{H}_k(\\mathbb{R}^n) \\subset L$\u3092\u793a\u305b\u3070\u3088\u3044. $L$\u306f\u30ce\u30eb\u30e0\u7a7a\u9593$\\mathcal{H}_k(\\mathbb{R}^n)$\u306e\u6709\u9650\u6b21\u5143\u90e8\u5206\u7a7a\u9593\u306a\u306e\u3067\u9589\u3067\u3042\u308b(\\cite{FujitaKurodaIto}\u306e{\\S}1.5 (a)\u53c2\u7167). \u3057\u305f\u304c\u3063\u3066, \u4efb\u610f\u306e$p \\in \\mathcal{H}_k(\\mathbb{R}^n)$\u306b\u5bfe\u3057\u3066, \u5c04\u5f71\u5b9a\u7406\u3088\u308a, $q \\in L , r \\in L^{\\perp}$\u304c\u5b58\u5728\u3057\u3066$p = q + r$\u3068\u306a\u308b. \u3068\u3053\u308d\u304c, $r(x) = \\sum c_{\\alpha} x^{\\alpha}$\u3068\u66f8\u304f\u3053\u3068\u306b\u3059\u308b\u3068, $r \\in L^{\\perp}$\u3088\u308a, \u4efb\u610f\u306e$d \\in L(\\Omega)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  0 \n  &= \\ip<r(x),(d^{\\mathrm{T}} x)^k> \\\\\n  &= r(\\partial) (d^{\\mathrm{T}} x)^k \\\\\n  &= \\sum_{|\\alpha|=k} c_{\\alpha} \\partial^{\\alpha} (d^{\\mathrm{T}} x)^k \\\\\n  &= \\sum_{|\\alpha|=k} c_{\\alpha} k! d^{\\alpha} \\\\\n  &= k! r(d)\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3086\u3048\u306b, $r(d) = 0 ~(\\forall d \\in L(\\Omega))$\u3067\u3042\u308b\u304b\u3089, \u4eee\u5b9a\u3088\u308a$r=0$\u3067\u3042\u308b. \u3088\u3063\u3066, $p = q \\in L$\u3067\u3042\u308b. \n\\end{proof}\n\\begin{thm}\n  $\\Omega \\subset \\mathrm{Mat}(n,m;\\mathbb{R})$\u3068\u3059\u308b. \n  \u3053\u306e\u3068\u304d, $\\mathrm{span}R(\\Omega)$\u304c$C(\\mathbb{R}^n)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3068, $L(\\Omega)$\u4e0a\u3067\u6052\u7b49\u7684\u306b$0$\u3092\u53d6\u308b$\\mathcal{P}(\\mathbb{R}^n)$\u306e\u5143\u304c$0$\u4ee5\u5916\u306b\u306a\u3044\u3053\u3068\u306f\u540c\u5024\u3067\u3042\u308b. \n\\end{thm}\n\\begin{proof}\n  $L(\\Omega)$\u4e0a\u3067$0$\u3092\u53d6\u308b$\\mathcal{P}(\\mathbb{R}^n)$\u306e\u5143\u306f$0$\u306b\u9650\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$k \\geq 0$\u306b\u5bfe\u3057\u3066$\\mathcal{H}_k(\\mathbb{R}^n) \\subset \\mathrm{span}R(\\Omega)$\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305d\u3046. \u3053\u308c\u304c\u793a\u3055\u308c\u308c\u3070, \u3059\u3079\u3066\u306e\u591a\u9805\u5f0f\u304c$\\mathrm{span}R(\\Omega)$\u306b\u542b\u307e\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u306e\u3067, Stone-Weierstrass\u306e\u5b9a\u7406\u306b\u3088\u308a$\\mathrm{span}R(\\Omega)$\u306f$C(\\mathbb{R}^n)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \u3055\u3066, \u3059\u3050\u4e0a\u306e\u88dc\u984c\u306b\u3088\u308a\n  \\[\n  \\mathcal{H}_k(\\mathbb{R}^n) = \\mathrm{span}\\{ \\mathbb{R}^n \\ni x \\mapsto (d^{\\mathrm{T}}x)^k \\in \\mathbb{R} \\mid d \\in L(\\Omega)\\}\n  \\]\n  \u3068\u306a\u308b. \u307e\u305f, $d \\in L(\\Omega)$\u306a\u3089\u3070$A \\in \\Omega$\u3068$y \\in \\mathbb{R}^m$\u304c\u5b58\u5728\u3057\u3066$d = A^{\\mathrm{T}}y$\u3068\u306a\u308b\u306e\u3067, $g(z) := (z^{T}y)^k$\u3068\u5b9a\u7fa9\u3059\u308c\u3070$g(Ax) = (A^{\\mathrm{T}}x)y)^k = (x^{\\mathrm{T}} Ay)^k = (x^{\\mathrm{T}} d)^k$\u3068\u306a\u308b. \u3088\u3063\u3066, $x \\mapsto (d^{\\mathrm{T}} x)^k$\u306f$R(\\Omega)$\u306b\u5c5e\u3059\u308b\u306e\u3067, \n  \\[\n   \\mathcal{H}_k(\\mathbb{R}^n) = \\mathrm{span}\\{ \\mathbb{R}^n \\ni x \\mapsto (d^{\\mathrm{T}}x)^k \\in \\mathbb{R} \\mid d \\in L(\\Omega)\\} \\subset \\mathrm{span}R(\\Omega)\n  \\]\n  \u3067\u3042\u308b. \u9006\u306e\u8a3c\u660e\u306b\u3064\u3044\u3066\u306fLin, Pinkus, 1992 \\cite{LinAndPinkus}\u3092\u53c2\u7167\u3055\u308c\u305f\u3044. \n\\end{proof}\n\u6700\u5f8c\u306b7\u7ae0\u3067\u4f7f\u3046\u547d\u984c\u3092\u8a3c\u660e\u3059\u308b. \u8a3c\u660e\u306b\u306f\u300c\u6589\u6b21\u591a\u9805\u5f0f\u306e\u6c7a\u5b9a\u554f\u984c\u300d\u306b\u95a2\u3059\u308b\u6b21\u306e\u4e8b\u5b9f\u3092\u4f7f\u3046. \n\\begin{prop}\\label{HomPolyDecis}\n  $n$\u5909\u6570$d$\u6b21\u6589\u6b21\u591a\u9805\u5f0f\u306f\u3042\u308b$r(n,d) := \\binom{n-1+d}{d}$\u500b\u306e\u76f8\u7570\u306a\u308b$0$\u3067\u306a\u3044\u70b9\u3067\u6c7a\u5b9a\u3055\u308c\u308b. \u3064\u307e\u308a, \u3042\u308b$\\xi^{(1)},\\ldots,\\xi^{(r(n,d))} \\in \\mathbb{R}^n\\setminus\\{0\\}$\u304c\u5b58\u5728\u3057\u3066, \u4efb\u610f\u306e$p \\in \\mathcal{H}_{d}(\\mathbb{R}^n)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  (\\forall i \\in \\{1,\\ldots,r(n,d) \\},~ p(\\xi^{(i)}) = 0) \\Rightarrow p \\equiv 0\n  \\]\n  \u3068\u306a\u308b. \u3055\u3089\u306b, \u3053\u306e\u3088\u3046\u306a$\\xi^{(1)},\\ldots,\\xi^{(r(n,d))} \\in \\mathbb{R}^n \\setminus \\{0\\}$\u3068\u3057\u3066, \u4efb\u610f\u306e$k = 1,\\ldots,d$\u3068$p \\in \\mathcal{H}_{k}(\\mathbb{R}^n)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  (\\forall i \\in \\{1,\\ldots,r(n,d) \\},~ p(\\xi^{(i)}) = 0) \\Rightarrow p \\equiv 0\n  \\]\n  \u3068\u306a\u308b\u3088\u3046\u306a\u3082\u306e\u304c\u53d6\u308c\u308b. \n\\end{prop}\n\\begin{proof}\n  \\cite{HomPolyDecision}\u3092\u53c2\u7167\u3055\u308c\u305f\u3044. \n\\end{proof}\n\\begin{prop}\\label{PolyRidgeRelation}\n  $r = \\mathrm{dim}\\mathcal{H}_k(\\mathbb{R}^n) = \\binom{n-1+k}{k}$\u3068\u304a\u304d, $n$\u5909\u6570\u306e$k$\u6b21\u4ee5\u4e0b\u306e\u591a\u9805\u5f0f\u5168\u4f53\u306e\u96c6\u5408\u3092$\\pi_k(\\mathbb{R}^n)$\u3068\u8868\u3059\u3053\u3068\u306b\u3059\u308b. \u3053\u306e\u3068\u304d, $a_1, \\ldots, a_r \\in \\mathbb{R}^n$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  \\pi_k(\\mathbb{R}^n) = \\left\\{ x \\mapsto \\sum_{i=1}^r g_i(a_i^{\\mathrm{T}} x )  \\mid g_i \\in \\pi_k(\\mathbb{R}) \\right\\}\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \n\\end{prop}\n\\begin{proof}\n  \u547d\u984c\\ref{HomPolyDecis}\u3088\u308a$a_1, \\ldots, a_r \\in \\mathbb{R}^n$\u304c\u5b58\u5728\u3057\u3066, \u5404$s = 0,\\ldots, k$\u306b\u5bfe\u3057\u3066, $A = \\{a_1,\\ldots,a_r\\}$\u4e0a\u3067\u6052\u7b49\u7684\u306b$0$\u3092\u53d6\u308b$\\mathcal{H}_s(\\mathbb{R}^n)$\u306e\u5143\u306f$0$\u306b\u9650\u308b. \u3057\u305f\u304c\u3063\u3066, \u88dc\u984c\\ref{HomPolyReprensent}\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\n  \\[\n  \\mathcal{H}_s(\\mathbb{R}^n) = \\mathrm{span}\\{ \\mathbb{R}^n \\ni x \\mapsto (d^{\\mathrm{T}}x)^s \\in \\mathbb{R} \\mid d \\in A\\}\n  \\]\n  \u3068\u306a\u308b. \u3053\u306e\u3053\u3068\u3068\u591a\u9805\u5f0f\u3092\u6589\u6b21\u591a\u9805\u5f0f\u306e\u548c\u3068\u3057\u3066\u8868\u305b\u308b\u3053\u3068\u3088\u308a\n  \\[\n  \\begin{aligned}\n  \\pi_k(\\mathbb{R}^n) \n  &= \\mathrm{span} \\{x \\mapsto (d^{\\mathrm{T}}x)^s \\mid d \\in A, s = 0,\\ldots, k\\} \\\\\n  &= \\left\\{ x \\mapsto \\sum_{i=1}^r g_i(a_i^{\\mathrm{T}} x )  \\mid g_i \\in \\pi_k(\\mathbb{R}) \\right\\}\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \n\\end{proof}\n\n\\section{\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e$C(\\mathbb{R}^r)$\u306b\u304a\u3051\u308b\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406}\n\u672c\u7ae0\u3067\u306f\u307e\u305a1993\u5e74\u306eLeshno et al., Hornik\u306e\u7d50\u679c\u3092\u8a3c\u660e\u3059\u308b. \n\u6b21\u3044\u3067, Cybenko\u306e\u7d50\u679c\u3092\u4e00\u822c\u5316\u3057\u305fChui\u3068Li\u306e\u7d50\u679c\u3092\u89e3\u8aac\u3059\u308b. \n\u3055\u3089\u306bChui\u3068Li\u306e\u7d50\u679c\u3092\u30ce\u30eb\u30e0\u7a7a\u9593\u3078\u4e00\u822c\u5316\u3057\u305f, Sun\u3068Cheney\u306b\u3088\u308b\u7d50\u679c\u3092\u7d39\u4ecb\u3057, \u305d\u306e\u5f8c, 1989\u5e74\u306eHornik\u306e\u7d50\u679c\u306e\u8a3c\u660e\u3092\u4e0e\u3048\u308b. \n\\subsection{Leshno et al.\u306e\u7d50\u679c}\n\u672c\u7bc0\u3067\u306f, Leshno et al.\u306e\u7d50\u679c\\cite{Leshno}\u304a\u3088\u3073\u305d\u306e\u6539\u826f\u7248\u306b\u3064\u3044\u3066\u8ff0\u3079\u308b. \u672c\u7bc0\u3067\u306f\u7279\u306b\u65ad\u3089\u306a\u3051\u308c\u3070a.e.\u306fLebesgue\u6e2c\u5ea6\u306e\u610f\u5473\u3067\u7528\u3044\u308b. \n\\begin{defn}\n  $\\Omega \\subset \\mathbb{R}^n$\u3092\u958b\u96c6\u5408\u3068\u3059\u308b. \u307e\u305f,  $\\mu$\u3092$(\\mathbb{R}^n,\\mathcal{B}_{\\mathbb{R}^n})$\u4e0a\u306eLebesgue\u6e2c\u5ea6\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \n  \\[\n  \\begin{aligned}\n  L_{\\mathrm{loc}}^{\\infty}(\\Omega) \n  &:= \\left\\{f:\\Omega \\rightarrow \\mathbb{R} ~\\vline~ \n  \\begin{aligned}\n  &\\mbox{\u4efb\u610f\u306e\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408}K \\subset \\Omega \\mbox{\u306b\u5bfe\u3057\u3066}\\\\\n  &f|_{K} \\in L^{\\infty}(K)\n  \\end{aligned} \n  \\right\\} \\\\\n  \\mathcal{M}(\\Omega) &:= \\{f \\in L_{\\mathrm{loc}}^{\\infty}(\\Omega) \\mid \\mu(\\mathrm{cl}\\{f\\mbox{\u306e\u4e0d\u9023\u7d9a\u70b9}\\}) = 0\\}\n  \\end{aligned}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \n\\end{defn}\nLeshno et al. \\cite{Leshno}\u3067\u793a\u3055\u308c\u305f\u4e3b\u8981\u306a\u7d50\u679c\u3092\u518d\u63b2\u3057\u3088\u3046. \n\\begin{thm}[Leshno et al. 1993 \\cite{Leshno}]\\label{LeshnoMainResult1} \\\\\n  $\\mu$\u3092$\\mathbb{R}$\u4e0a\u306eLebesgue\u6e2c\u5ea6\u3068\u3059\u308b. $\\Psi \\in \\mathcal{M}(\\mathbb{R})$\u306b\u5bfe\u3057\u3066, $\\Sigma^r(\\Psi)$\u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u305f\u3081\u306e\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306f, \u3044\u304b\u306a\u308b\u4e00\u5909\u6570\u591a\u9805\u5f0f\u95a2\u6570$P:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306b\u3064\u3044\u3066\u3082$\\Psi = P ~(\\mu \\mathrm{-a.e.})$\u3068\u306a\u3089\u306a\u3044\u3053\u3068\u3067\u3042\u308b. \n\\end{thm}\n\u547d\u984c\\ref{DensityDim1Suff}\u3088\u308a\u3053\u306e\u5b9a\u7406\u3092\u793a\u3059\u306b\u306f$r=1$\u306e\u5834\u5408\u3092\u793a\u305b\u3070\u3088\u3044. \u307e\u305a\u5fc5\u8981\u6027\u3092\u793a\u305d\u3046. \u5fc5\u8981\u6027\u306e\u8a3c\u660e\u306f\u975e\u5e38\u306b\u5358\u7d14\u3067\u3042\u308b. \n\\begin{lem}\n  \u95a2\u6570$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u304ca.e.\u3067\u591a\u9805\u5f0f\u95a2\u6570\u3067\u3042\u308b\u306a\u3089\u3070, $\\Sigma^1(\\Psi)$\u306f$C(\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u306a\u3044. \n\\end{lem}\n\\begin{proof}\n  $\\mathrm{deg}(\\Psi) = n$\u3068\u3059\u308b\u3068, $\\Sigma^r(\\Psi)$\u306f$n$\u6b21\u306e$1$\u5909\u6570\u591a\u9805\u5f0f\u3068a.e.\u3067\u4e00\u81f4\u3059\u308b\u95a2\u6570\u5168\u4f53\u306e\u96c6\u5408\u306b\u542b\u307e\u308c\u308b. \u3057\u305f\u304c\u3063\u3066, $n+1$\u6b21\u306e$1$\u5909\u6570\u591a\u9805\u5f0f\u306f\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u8fd1\u4f3c\u3067\u304d\u306a\u3044. \n\\end{proof}\n\u5341\u5206\u6027\u306f\u6b21\u306e\u5b9a\u7406\u306b\u542b\u307e\u308c\u308b\u305f\u3081\u672c\u7bc0\u3067\u306f\u4ee3\u308f\u308a\u306b\u6b21\u306e\u5b9a\u7406\u3092\u8a3c\u660e\u3059\u308b. \n\\begin{thm}\n  $W \\subset \\mathbb{R}$\u306f\u5b64\u7acb\u70b9\u3092\u6301\u305f\u305a$0 \\in W$\u3067\u3042\u308b\u3068\u3059\u308b. \n  \u307e\u305f, $\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f$\\ \\mathcal{M}(\\mathbb{R})$\u306b\u5c5e\u3059\u308b\u3068\u3059\u308b. \u3055\u3089\u306b, \u3042\u308b\u958b\u533a\u9593$I$\u306b\u5bfe\u3057\u3066$\\Psi$\u304c$I$\u4e0aa.e.\u306b\u591a\u9805\u5f0f\u3068\u4e00\u81f4\u3057\u306a\u3044\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $\\mathcal{A} \\subset \\mathbb{R}^r$\u4e0a\u3067$0$\u3092\u53d6\u308b\u975e\u81ea\u660e\u306a$r$\u5909\u6570\u6589\u6b21\u591a\u9805\u5f0f\u304c\u5b58\u5728\u3057\u306a\u3044\u306a\u3089\u3070, \n  \\[\n  \\begin{aligned}\n  \\mathcal{N}_r(\\Psi,W\\mathcal{A},I) = \\mathrm{span}\\{ \\mathbb{R}^r \\ni x \\mapsto \\Psi(w a^{T} x + b) \\in \\mathbb{R} \\mid w \\in W, a \\in \\mathcal{A}, b \\in I \\}\n  \\end{aligned}\n  \\]\n  \u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{thm}\n\u3055\u3089\u306b\u3053\u306e\u5b9a\u7406\u306f\u547d\u984c\\ref{MultiVariateDensity}\u3068\u6b21\u306e\u5b9a\u7406\u3088\u308a\u5f93\u3046\u306e\u3067\u6b21\u306e\u5b9a\u7406\u3092\u8a3c\u660e\u3059\u308c\u3070\u3088\u3044. \n\\begin{thm*}\n  $W \\subset \\mathbb{R}$\u306f\u5b64\u7acb\u70b9\u3092\u6301\u305f\u305a$0 \\in W$\u3067\u3042\u308b\u3068\u3059\u308b. \n  \u307e\u305f, $\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f$\\mathcal{M}(\\mathbb{R})$\u306b\u5c5e\u3059\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u958b\u533a\u9593$I$\u306b\u5bfe\u3057\u3066$\\Psi$\u304c$I$\u4e0aa.e.\u306b\u591a\u9805\u5f0f\u3068\u4e00\u81f4\u3057\u306a\u3044\u306a\u3089, $\\mathcal{N}_1(\\Psi,W,I)$\u306f$C(\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{thm*}\n\u307e\u305a$\\Psi$\u304c$C^{\\infty}$\u7d1a\u3067\u3042\u308b\u5834\u5408\u306b\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u8a3c\u660e\u3057\u3088\u3046. \n\\begin{lem}\\label{NeuralNetApproPoly}\n  $W \\subset \\mathbb{R}$\u306f\u5b64\u7acb\u70b9\u3092\u6301\u305f\u305a$0 \\in W$\u3067\u3042\u308b\u3068\u3059\u308b. \n  \u307e\u305f, $\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f\u3042\u308b\u958b\u533a\u9593$I$\u4e0a\u3067$C^{\\infty}$\u304b\u3064\u591a\u9805\u5f0f\u3068\u4e00\u81f4\u3057\u306a\u3044\u3068\u3059\u308b. \n  \u3053\u306e\u3068\u304d, \u5404\u6574\u6570$n \\geq 0$\u306b\u3064\u3044\u3066\u591a\u9805\u5f0f$x \\mapsto x^n$\u306f$\\mathcal{N}_1(\\Psi,W,I)$\u3067\u5e83\u7fa9\u4e00\u69d8\u8fd1\u4f3c\u3055\u308c\u308b. \u3057\u305f\u304c\u3063\u3066, \u7279\u306b,  $\\mathcal{N}_1(\\Psi,W,I)$\u306f$C(\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{lem}\n\\begin{proof}\n  \u5f8c\u534a\u306b\u3064\u3044\u3066\u306f$C(\\mathbb{R})$\u306e\u5143\u304c\u591a\u9805\u5f0f\u95a2\u6570\u3067\u5e83\u7fa9\u4e00\u69d8\u8fd1\u4f3c\u3067\u304d\u308b\u3053\u3068, \u304a\u3088\u3073\u524d\u534a\u3088\u308a$\\mathcal{N}_1(\\Psi,W,I)$\u304c\u5168\u3066\u306e$1$\u5909\u6570\u306e$\\mathbb{R}$\u4fc2\u6570\u591a\u9805\u5f0f\u3092\u5e83\u7fa9\u4e00\u69d8\u8fd1\u4f3c\u3067\u304d\u308b\u3053\u3068\u304b\u3089\u308f\u304b\u308b. \n  $K \\subset \\mathbb{R}$\u3092\u7a7a\u3067\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u3068\u3059\u308b. \u307e\u305f, $b \\in I$\u306b\u5bfe\u3057\u3066, $f_b:\\mathbb{R} \\times \\mathbb{R} \\ni (x,w) \\mapsto \\Psi(wx + b) \\in \\mathbb{R}$\u3068\u5b9a\u7fa9\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$k \\in \\mathbb{N}, w_0 \\in W, b \\in I$\u306b\u5bfe\u3057\u3066, \u95a2\u6570$x \\mapsto ((\\partial w)^k f_b) (x,w_0)$\u304c$\\mathcal{N}_1(\\Psi,W,I)$\u306e\u5143\u3067$K$\u4e0a\u4e00\u69d8\u8fd1\u4f3c\u3067\u304d\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u5341\u5206\u3067\u3042\u308b. \u306a\u305c\u306a\u3089, \u3082\u3057\u305d\u308c\u304c\u793a\u3055\u308c\u308c\u3070, $\\Psi$\u304c$I$\u4e0a\u3067\u591a\u9805\u5f0f\u95a2\u6570\u3067\u306a\u3044\u3053\u3068\u304b\u3089,  \u4efb\u610f\u306e$k$\u306b\u3064\u3044\u3066$\\Psi^{(k)}(b_k) \\neq 0$\u306a\u308b$b_k \\in I$\u304c\u53d6\u308c\u308b\u306e\u3067, $w_0=0, b = b_k$\u3068\u3059\u308b\u3053\u3068\u3067, \u95a2\u6570$x \\mapsto ((\\partial w)^k f_{b_k}) (x,w_0) = x^k \\Psi^{(k)}(w_0 x + b_k) = x^k \\Psi^{(k)}(b_k)$\u304c$\\mathcal{N}_1(\\Psi,W,I)$\u306e\u5143\u3067$K$\u4e0a\u4e00\u69d8\u8fd1\u4f3c\u3067\u304d\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u3055\u3066, \u307e\u305a$k=1$\u306e\u5834\u5408\u3092\u793a\u305d\u3046.  \u4efb\u610f\u306b$\\delta_0,\\varepsilon > 0$\u3092\u53d6\u308b. \u3044\u307e, \n  \\[\n  K \\times [w_0-\\delta_0,w_0+\\delta_0] \\ni (x,w) \\mapsto ((\\partial w) f_b)(x,w) \\in \\mathbb{R}\n  \\]\n  \u306f\u9023\u7d9a\u3067\u3042\u308b. $K \\times [w_0-\\delta_0,w_0+\\delta_0]$\u306f\u30b3\u30f3\u30d1\u30af\u30c8\u3060\u304b\u3089, $0 < \\delta < \\delta_0$\u304c\u5b58\u5728\u3057\u3066, \u4efb\u610f\u306e$(x_1,w_1),(x_2,w_2) \\in K \\times [w_0-\\delta_0,w_0+\\delta_0]$\u306b\u3064\u3044\u3066, $|x_1 - x_2|,|w_1-w_2| < \\delta$\u306a\u3089\u3070, $|((\\partial w) f_b)(x_1,w_1) - ((\\partial w) f_b)(x_2,w_2)| < \\varepsilon$\u3068\u306a\u308b. \u305d\u3053\u3067, $-\\delta < h < \\delta$\u304b\u3064$w_0+h \\in W$\u3068\u306a\u308b\u3088\u3046\u306b$h$\u3092\u3068\u308b\u3068($W$\u306f\u5b64\u7acb\u70b9\u3092\u6301\u305f\u306a\u3044\u306e\u3067\u3068\u308c\u308b),  \u4efb\u610f\u306e$x \\in K$\u306b\u5bfe\u3057\u3066, \u5e73\u5747\u5024\u306e\u5b9a\u7406\u3088\u308a, \n  \\[\n  \\begin{aligned}\n  &\\left|\\frac{\\Psi((w_0+h)x+b) - \\Psi(w_0 x + b)}{h} -  ((\\partial w) f_b)(x,w_0) \\right| \\\\\n  &\\left|\\frac{f_b(x,w_0+h) - f_b(x,w_0)}{h} -  ((\\partial w) f_b)(x,w_0) \\right| \\\\\n  &= |((\\partial w) f_b)(x,w_0+\\theta h) - ((\\partial w) f_b)(x,w_0)| ~~~(\\exists \\theta \\in [0,1]) \\\\\n  &\\leq \\varepsilon\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3053\u308c\u3067$k=1$\u306e\u5834\u5408\u304c\u8a3c\u660e\u3055\u308c\u305f.\n  \u6b21\u306b$k$\u3067\u6210\u7acb\u3059\u308b\u3068\u3059\u308b\u3068\u3057\u3066$k+1$\u3067\u306e\u6210\u7acb\u3092\u793a\u305d\u3046. \n  $k=1$\u306e\u5834\u5408\u306e\u8a3c\u660e\u3068\u540c\u69d8\u306b\u3057\u3066, \u4efb\u610f\u306e$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066\u3042\u308b$h \\in \\mathbb{R}$\u304c\u5b58\u5728\u3057\u3066, $w_0+h \\in W$\u304b\u3064\u4efb\u610f\u306e$x \\in K$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\left| \\frac{((\\partial w)^k f_b)(x,w_0+h) - ((\\partial w)^k f_b)(x,w_0)}{h} - ((\\partial w)^{k+1} f_b)(x,w_0)\\right| < \\varepsilon\n  \\]\n  \u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u5e30\u7d0d\u6cd5\u306e\u4eee\u5b9a\u304b\u3089, \n  \\[\n  \\begin{aligned}\n  &x \\mapsto ((\\partial w)^k f_b)(x,w_0+h)\\\\\n  &x \\mapsto ((\\partial w)^k f_b)(\\cdot,w_0)\n  \\end{aligned}\n  \\]\n  \u306f$\\mathcal{N}_1(\\Psi,W,I)$\u3067$K$\u4e0a\u4e00\u69d8\u8fd1\u4f3c\u3067\u304d\u308b. \u3086\u3048\u306b, \u4e0a\u5f0f\u3068\u5408\u308f\u305b\u308b\u3068\\\\\n  $((\\partial w)^{k+1} f_b)(\\cdot,w_0)$\u306f$\\mathcal{N}_1(\\Psi,W,I)$\u3067$K$\u4e0a\u4e00\u69d8\u8fd1\u4f3c\u3055\u308c\u308b. \n  \u3053\u308c\u3067\u793a\u305b\u305f. \n\\end{proof}\n\n\u6b21\u306b\u4e00\u822c\u306e$\\Psi$\u306e\u5834\u5408\u3092\u8a3c\u660e\u3059\u308b\u305f\u3081\u306b\u3044\u304f\u3064\u304b\u88dc\u984c\u3092\u793a\u3059. \n\\begin{lem}\n  $\\Psi \\in \\mathcal{M}(\\mathbb{R})$\u3068\u3057, $\\mu$\u3092$\\mathbb{R}$\u4e0a\u306eLebesgue\u6e2c\u5ea6\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $a \\leq b$\u3068$\\delta > 0$\u306b\u5bfe\u3057\u3066\u6709\u9650\u500b\u306e\u958b\u533a\u9593$I_1, \\ldots, I_n $\u304c\u5b58\u5728\u3057\u3066, $U = \\bigcup_{k=1}^n I_k$\u306b\u3064\u3044\u3066, $\\mu(U) < \\delta$\u304b\u3064$\\Psi$\u306f$[a,b] \\setminus U$\u4e0a\u3067\u4e00\u69d8\u9023\u7d9a\u3067\u3042\u308b. \n\\end{lem}\n\\begin{proof}\n  $A := [a,b] \\cap \\mathrm{cl}\\{\\Psi \\mbox{\u306e\u4e0d\u9023\u7d9a\u70b9}\\}$\u3068\u304a\u304f. $\\Psi \\in \\mathcal{M}(\\mathbb{R})$\u306a\u306e\u3067$\\mu(A) = 0$\u3067\u3042\u308b. \u3059\u308b\u3068, Lebesgue\u6e2c\u5ea6\u306e\u5916\u6b63\u5247\u6027\u3088\u308a, \u958b\u96c6\u5408$U \\supset A$\u3067$\\mu(U) < \\delta$\u306a\u308b\u3082\u306e\u304c\u53d6\u308c\u308b. $\\mathbb{R}$\u306e\u958b\u96c6\u5408\u306f\u958b\u533a\u9593\u306e\u548c\u96c6\u5408\u3067\u8868\u305b\u308b\u306e\u3067, $U = \\bigcup_{j \\in J}I_j$\u3068\u8868\u3059\u3053\u3068\u306b\u3059\u308b\u3068, $A$\u306e\u30b3\u30f3\u30d1\u30af\u30c8\u6027\u3088\u308a$j_1,\\ldots,j_m \\in J$\u304c\u5b58\u5728\u3057\u3066$A \\subset \\bigcup_{k=1}^{m} I_{j_k}$\u3068\u306a\u308b. $I = \\bigcup_{k=1}^{m} I_{j_k}$\u3068\u304a\u304f\u3068$[a,b] \\setminus I$\u306f$\\Psi$\u306e\u4e0d\u9023\u7d9a\u70b9\u3092\u542b\u307e\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u3067\u3042\u308b.  \u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u4e0a\u306e\u9023\u7d9a\u95a2\u6570\u306f\u4e00\u69d8\u9023\u7d9a\u3067\u3042\u308b\u306e\u3067 $I_{j_1},\\ldots,I_{j_m}$\u304c\u6c42\u3081\u308b\u3082\u306e\u3067\u3042\u308b. \n\\end{proof}\n\\begin{lem}\\label{CovolutionUniformAppro}\n  $\\Psi \\in \\mathcal{M}(\\mathbb{R})$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$\\rho \\in C_0^{\\infty}(\\mathbb{R})$\u306b\u5bfe\u3057\u3066, $\\Psi$\u3068$\\rho$\u306e\u7573\u307f\u8fbc\u307f\u7a4d\n  \\[\n  (\\Psi * \\rho)(x) = \\int_{\\mathbb{R}} \\Psi(x-y) \\rho(y) dy\n  \\]\n  \u306f$\\mathcal{N}_1(\\Psi,\\{1\\},\\mathbb{R})$\u306b\u3088\u308a\u5e83\u7fa9\u4e00\u69d8\u8fd1\u4f3c\u3055\u308c\u308b. \n\\end{lem}\n\\begin{proof}\n  \u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}$\u3092\u3068\u308b. $\\alpha> 0$\u3092\u9069\u5f53\u306b\u53d6\u308a$K$\u3068$\\mathrm{supp}(\\rho)$\u304c$ [-\\alpha,\\alpha]$\u306b\u542b\u307e\u308c\u308b\u3088\u3046\u306b\u3059\u308b.  $[-\\alpha,\\alpha]$\u4e0a\u3067\u4e00\u69d8\u8fd1\u4f3c\u3067\u304d\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u5341\u5206\u3067\u3042\u308b. \u305d\u3053\u3067, $m \\in \\mathbb{N}$\u306b\u5bfe\u3057\u3066$y_j = - \\alpha + 2j\\alpha\/m, \\Delta y_i = 2 \\alpha\/m ~(i=1,\\ldots,m)$\u3068\u304a\u304d, \n  \\[\n  x \\mapsto \\sum_{j=1}^{m} \\Psi(x - y_j) \\rho(y_j) \\Delta y_j\n  \\]\n  \u304c$\\Psi * \\rho$\u306b$[-\\alpha,\\alpha]$\u4e0a\u4e00\u69d8\u53ce\u675f\u3059\u308b\u3053\u3068\u3092\u793a\u305d\u3046. $\\rho \\neq 0$\u3068\u3057\u3066\u3088\u3044. $\\varepsilon > 0$\u3092\u4efb\u610f\u306b\u53d6\u308b. $0 < \\delta < \\alpha$\u3092\u6b21\u3092\u307f\u305f\u3059\u3088\u3046\u306b\u53d6\u308b. \n  \\[\n  10 \\delta \\lVert \\Psi \\rVert_{L^{\\infty}([-\\alpha,\\alpha])} \\lVert \\rho \\rVert_{L^{\\infty}} \\leq \\varepsilon.\n  \\]\n  \u3044\u307e, $\\Psi \\in \\mathcal{M}(\\mathbb{R})$\u306a\u306e\u3067, $r(\\delta)$\u500b\u306e\u958b\u533a\u9593\u304c\u5b58\u5728\u3057\u3066, \u305d\u308c\u3089\u306e\u548c\u96c6\u5408\u3092$U$\u3068\u3059\u308b\u3068, \u305d\u306eLebesgue\u6e2c\u5ea6$\\mu(U)$\u306f$\\mu(U) < \\delta$\u3092\u307f\u305f\u3057, \u304b\u3064$\\Psi$\u306f$[-2\\alpha,2\\alpha] \\setminus U$\u4e0a\u3067\u4e00\u69d8\u9023\u7d9a\u3068\u306a\u308b. \u305d\u3057\u3066, $m \\in \\mathbb{N}$\u3092$m \\delta > \\alpha r(\\delta)$\u304b\u3064, \n  \\[\n  \\begin{aligned}\n    &|s-t| < 2\\alpha\/m \\Rightarrow |\\rho(s) - \\rho(t)| \\leq \\frac{\\varepsilon}{2\\alpha \\lVert \\Psi \\rVert_{L^{\\infty}([-2\\alpha,2\\alpha])}} \\\\\n    &s,t \\in [-2\\alpha,2\\alpha] \\setminus U, |s-t| < 2\\alpha\/m \\Rightarrow |\\Psi(s) - \\Psi(t)| \\leq \\frac{\\varepsilon}{\\lVert \\rho \\rVert_{L^1}}\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b\u3088\u3046\u306b\u53d6\u308b($\\rho$\u306f\u4e00\u69d8\u9023\u7d9a\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f). \u3053\u3053\u3067$x \\in [-\\alpha,\\alpha]$\u3092\u56fa\u5b9a\u3059\u308b. $\\Delta_j = [y_{j-1},y_j]$\u3068\u304a\u304f(\u305f\u3060\u3057, $y_0 = \\alpha$\u3068\u3059\u308b)\u3068, $\\mathrm{supp}(\\rho) \\subset [-\\alpha,\\alpha]$\u3067\u3042\u308b\u304b\u3089, \n  \\[\n  \\begin{aligned}\n  &\\left| \\int_{\\mathbb{R}} \\Psi(x-y) \\rho(y) dy - \\sum_{j=1}^m \\int_{\\Delta_j} \\Psi(x - y_j) \\rho(y) dy \\right| \\\\\n  &\\leq \\sum_{j=1}^m \\int_{\\Delta_j} |\\Psi(x-y) - \\Psi(x-y_j)||\\rho(y)| dy\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. $y \\in \\Delta_j$\u306b\u3064\u3044\u3066$|y - y_j| < 2\\alpha\/m$\u3067\u3042\u308a, $x - y \\in [-2\\alpha,2\\alpha]$\u3067\u3042\u308b\u306e\u3067, \u3082\u3057$(x - \\Delta_j) \\cap U = \\emptyset$\u3067\u3042\u308b\u306a\u3089\u3070, \n  \\[\n  \\int_{\\Delta_j} |\\Psi(x-y) - \\Psi(x-y_j)||\\rho(y)| dy \\leq \\frac{\\varepsilon}{\\lVert \\rho \\rVert_{L^1}} \\int_{\\Delta_j} |\\rho(y)| dy\n  \\]\n  \u3068\u306a\u308b. \u3086\u3048\u306b, \u305d\u306e\u3088\u3046\u306a$j$\u3059\u3079\u3066\u306b\u3064\u3044\u3066\u548c\u3092\u53d6\u308b\u3068, \u305d\u306e\u548c\u306f$\\varepsilon$\u3067\u6291\u3048\u3089\u308c\u308b. \u307e\u305f, $(x - \\Delta_j) \\cap U \\neq \\emptyset$\u3068\u306a\u308b$\\Delta_j$\u3092$\\tilde{\\Delta}_j$\u3067\u304b\u304f\u3053\u3068\u306b\u3059\u308b\u3068, $U$\u306eLebesgue\u6e2c\u5ea6\u304c$\\delta$\u672a\u6e80\u3067\u3042\u308b\u3053\u3068\u53ca\u3073 $U$\u304c$r(\\delta)$\u500b\u306e\u958b\u533a\u9593\u306e\u548c\u96c6\u5408\u3067\u3042\u308b\u3053\u3068\u304b\u3089,  $\\tilde{\\Delta}_j$\u306eLebesgue\u6e2c\u5ea6\u306e\u7dcf\u548c\u306f$\\delta + (4\\alpha\/m)r(\\delta)$\u3067\u6291\u3048\u3089\u308c\u308b. \u3057\u305f\u304c\u3063\u3066, $m$\u306e\u9078\u3073\u65b9\u304b\u3089$\\tilde{\\Delta}_j$\u306eLebesgue\u6e2c\u5ea6\u306e\u7dcf\u548c\u306f$5 \\delta$\u3067\u6291\u3048\u3089\u308c\u308b. \u3088\u3063\u3066, $\\delta$\u306e\u53d6\u308a\u65b9\u304b\u3089, \n  \\[\n  \\sum \\int_{\\tilde{\\Delta}_i} |\\Psi(x-y) - \\Psi(x-y_j)||\\rho(y)| dy \\leq 2\\lVert \\Psi \\rVert_{L^{\\infty}([-2\\alpha,2\\alpha])} \\lVert \\rho \\rVert_{L^{\\infty}} 5 \\delta \\leq \\varepsilon\n  \\]\n  \u3068\u306a\u308b. \u6700\u5f8c\u306b, $\\rho$\u306b\u5bfe\u3059\u308b$m$\u306e\u53d6\u308a\u65b9\u304b\u3089, \n  \\[\n  \\begin{aligned}\n  &\\left| \\sum_{j=1}^m \\int_{\\Delta_j} \\Psi(x - y_j) \\rho(y) dy - \\sum_{j=1}^m \\Psi(x-y_j)\\rho(y_j)\\Delta y_j \\right| \\\\\n  &= \\left| \\sum_{j=1}^m \\int_{\\Delta_j} \\Psi(x - y_j) ( \\rho(y)  - \\rho(y_j))dy \\right| \\\\\n  &\\leq \\sum_{j=1}^m  \\int_{\\Delta_j} |\\Psi(x - y_j)|~| \\rho(y)  - \\rho(y_j)| dy \\\\\n  &\\leq \\sum_{j=1}^m  \\int_{\\Delta_j} \\lVert \\Psi \\rVert_{L^{\\infty}([-2\\alpha,2\\alpha])} \\frac{\\varepsilon}{2\\alpha \\lVert \\Psi \\rVert_{L^{\\infty}([-2\\alpha,2\\alpha])}} dy \\\\\n  &= \\varepsilon\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u4ee5\u4e0a\u304b\u3089, \u4efb\u610f\u306e$x \\in [-\\alpha,\\alpha]$\u306b\u5bfe\u3057\u3066, \u4e09\u89d2\u4e0d\u7b49\u5f0f\u306b\u3088\u308a, \n  \\[\n  \\left| \\int_{\\mathbb{R}} \\Psi(x-y) \\rho(y) dy - \\sum_{j=1}^m \\Psi(x-y_j) \\rho(y_j) \\Delta y_j \\right| \\leq 3 \\varepsilon.\n  \\]\n\\end{proof}\n\\begin{rem}\n  \u4e0a\u306e\u5b9a\u7406\u3067\u300c$\\mathcal{N}_1(\\Psi,\\{1\\},\\mathbb{R})$\u306b\u3088\u308a\u5e83\u7fa9\u4e00\u69d8\u8fd1\u4f3c\u3055\u308c\u308b\u300d\u3068\u8ff0\u3079\u305f\u304c, $y_j \\notin \\mathrm{supp}(\\rho)$\u306e\u3068\u304d$\\Psi(x - y_j) \\rho(y_j) = 0$\u3067\u3042\u308b\u306e\u3067,  $\\mathcal{N}_1(\\Psi,\\{1\\},\\mathrm{supp}(\\rho))$\u306b\u3088\u308a\u5e83\u7fa9\u4e00\u69d8\u8fd1\u4f3c\u3055\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \n\\end{rem}\n\\begin{lem}\\label{PointwiseConvergencePolyPreserve}\n  $N \\in \\mathbb{N}$\u3068\u3059\u308b. $p_j:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3092$\\mathrm{deg}(p_j) \\leq N$\u306a\u308b\u591a\u9805\u5f0f\u95a2\u6570\u306e\u5217\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306b\u5bfe\u3057\u3066, $p_j$\u304c$f$\u306b\u5404\u70b9\u53ce\u675f\u3059\u308b\u306a\u3089\u3070, $f$\u306f\u3042\u308b$N$\u6b21\u4ee5\u4e0b\u306e\u591a\u9805\u5f0f\u3068\u4e00\u81f4\u3059\u308b. \u307e\u305f, $p_j$\u304c$f$\u306b\u6982\u53ce\u675f\u3059\u308b(\u3064\u307e\u308a, a.e.$x \\in \\mathbb{R}$\u306b\u5bfe\u3057\u3066$p_j(x) \\rightarrow f(x)$\u3068\u306a\u308b)\u306a\u3089\u3070, $f$\u306f\u3042\u308b$N$\u6b21\u4ee5\u4e0b\u306e\u591a\u9805\u5f0f\u3068a.e.\u3067\u4e00\u81f4\u3059\u308b.\n\\end{lem}\n\\begin{proof}\n  \u307e\u305a, \u5404\u70b9\u53ce\u675f\u306e\u5834\u5408\u3092\u8003\u3048\u308b. $p_j(x) = \\sum_{k=0}^N a_{j,k} x^k$\u3068\u8868\u3059\u3053\u3068\u306b\u3059\u308b. \u76f8\u7570\u306a\u308b$x_0, x_1,\\ldots, x_N \\in \\mathbb{R} $\u3092\u4efb\u610f\u306b\u53d6\u308a, \n  \\[\n  \\begin{aligned}\n  &\\mathbf{p}^{(j)}\n  =\n  \\left(\n  \\begin{array}{c}\n       p_j(x_0)  \\\\\n       p_j(x_1) \\\\\n       \\vdots \\\\\n       p_j(x_N)\n  \\end{array}\n  \\right)\n  , ~~~\n  \\mathbf{X}\n  =\n  \\left(\n  \\begin{array}{cccc}\n    1 & x_0^1 & \\cdots & x_0^N  \\\\\n    1 & x_1^1 & \\cdots & x_1^N \\\\\n    \\vdots & \\vdots & \\ddots & \\vdots \\\\\n    1 & x_N^1 & \\cdots & x_N^N\n  \\end{array}\n  \\right)\n  , \\\\\n  &\\mathbf{a}^{(j)}\n  =\n  \\left(\n  \\begin{array}{c}\n    a_{j,0}  \\\\\n    a_{j,1} \\\\\n    \\vdots \\\\\n    a_{j,N}\n  \\end{array}\n  \\right)\n  , ~~~\n  \\mathbf{f}\n  =\n  \\left(\n  \\begin{array}{c}\n    f(x_0) \\\\\n    f(x_1) \\\\\n    \\vdots \\\\\n    f(x_N)\n  \\end{array}\n  \\right)\n  \\end{aligned}\n  \\]\n  \u3068\u304a\u304f. \u3059\u308b\u3068, Vandermonde\u884c\u5217$\\mathbf{X}$\u306f\u6b63\u5247\u3067\u3042\u308a, \n  \\[\n  \\mathbf{p}^{(j)}-\\mathbf{f} = \\mathbf{X} \\mathbf{a}^{(j)} - \\mathbf{f}\n  \\]\n  \u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, $p_j(x_i) \\rightarrow f(x_i)$\u3067\u3042\u308b\u3053\u3068\u3068\u884c\u5217\u5199\u50cf\u304c\u9023\u7d9a\u3067\u3042\u308b\u3053\u3068\u304b\u3089, \n  \\[\n  \\mathbf{a}^{(j)} = \\mathbf{X}^{-1} \\mathbf{f} + \\mathbf{X}^{-1} (\\mathbf{p}^{(j)}-\\mathbf{f}) \\rightarrow \\mathbf{X}^{-1} \\mathbf{f} ~~(j \\rightarrow \\infty)\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, $(a_1,\\ldots,a_N)^{\\mathrm{T}} = \\mathbf{X}^{-1} \\mathbf{f}$\u3068\u304a\u304d, $p(x) = \\sum_{k=0}^N a_k x^k$\u3068\u304a\u304f\u3068, \u5404\u70b9\u3067$p_j \\rightarrow p$\u3068\u306a\u308b. \u3088\u3063\u3066, $f = p$\u3068\u306a\u308b.  \u6982\u53ce\u675f\u306e\u5834\u5408\u3082\u540c\u69d8\u306b\u793a\u305b\u308b(Lebesgue\u6e2c\u5ea6\u3067\u8003\u3048\u3066\u3044\u308b\u306e\u3067\u53ce\u675f\u70b9\u3092$N+1$\u500b\u53d6\u308c\u308b\u3053\u3068\u306b\u6ce8\u610f). \n\\end{proof}\n\\begin{lem}\\label{ConvolutionAlmostEverywhereConvergence}\n  $\\Psi \\in \\mathcal{M}(\\mathbb{R})$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $\\rho_j \\in C_0^{\\infty}(\\mathbb{R}) ~(j=1,2,\\ldots)$\u304c\u5b58\u5728\u3057\u3066, $\\mathrm{supp}(\\rho_j) \\subset [-1\/j,1\/j]$\u304b\u3064$\\Psi * \\rho_j$\u306f$\\Psi$\u306b$\\mathbb{R}$\u4e0a\u6982\u53ce\u675f\u3059\u308b. \n\\end{lem}\n\\begin{proof}\n  $\\rho \\in C_0^{\\infty}(\\mathbb{R})$\u3067$\\int_{\\mathbb{R}} \\rho(x)dx = 1$\u304b\u3064$\\mathrm{supp}(\\rho) \\subset [-1,1]$\u3092\u307f\u305f\u3059\u3082\u306e\u3092\u3068\u308b. \u4f8b\u3048\u3070, \n  \\[\n  k(t) := \\left\\{\\begin{array}{cl}e^{-\\frac{1}{t}}&(t>0)\\\\0&(t\\leq0)\\end{array}\\right.\n  \\]\n  \u3068\u304a\u304d, $H(x) := k(1-|x|), ~\\rho(x) := H(x)\/\\int_{\\mathbb{R}} H(y) dy$\u3068\u304a\u3051\u3070\u3053\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059($k \\in C^\\infty(\\mathbb{R})$\u306e\u4e01\u5be7\u306a\u8a3c\u660e\u304c\u677e\u672c\\cite{MatsumotoManifold}\u306ep.176\u301c178\u306b\u3042\u308b\u306e\u3067\u53c2\u7167\u3055\u308c\u305f\u3044). \u305d\u3057\u3066, \n  \\[\n  \\rho_j(x) := j \\rho(jx) ~~(x \\in \\mathbb{R}, j \\in \\mathbb{N})\n  \\]\n  \u3068\u304a\u304f. \u3059\u308b\u3068, $\\mathrm{supp}(\\rho_j) \\subset [-1\/j,1\/j]$\u3086\u3048$\\rho_j \\in C_0^{\\infty}(\\mathbb{R})$\u3067\u3042\u308a, $\\Psi$\u306e\u9023\u7d9a\u70b9$x \\in \\mathbb{R}$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  (\\Psi*\\rho_j)(x) \n  &= \\int_{\\mathbb{R}} \\Psi(x-y)\\rho_j(y) dy \\\\\n  &= \\int_{\\mathbb{R}} \\Psi(x-z\/j) \\rho(z)dz ~~~~(z = jy) \\\\\n  &\\rightarrow \\int_{\\mathbb{R}} \\Psi(x) \\rho(z)dz = \\Psi(x) ~~~~(\\mbox{\u512a\u53ce\u675f\u5b9a\u7406})\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. $\\Psi$\u306b\u95a2\u3059\u308b\u4eee\u5b9a\u3088\u308a$\\Psi$\u306e\u4e0d\u9023\u7d9a\u70b9\u5168\u4f53\u306e\u96c6\u5408\u306e\u6e2c\u5ea6\u306f$0$\u306a\u306e\u3067\u3053\u308c\u3067\u8a3c\u660e\u3067\u304d\u305f. \n\\end{proof}\n\u4ee5\u4e0a\u3067\u6e96\u5099\u304c\u6574\u3063\u305f\u306e\u3067\u5b9a\u7406\u3092\u8a3c\u660e\u3057\u3066\u3044\u3053\u3046. \n\\begin{thm}\\label{GeneByPinkus}\n  $W \\subset \\mathbb{R}$\u306f\u5b64\u7acb\u70b9\u3092\u6301\u305f\u305a$0 \\in W$\u3067\u3042\u308b\u3068\u3059\u308b. \n  \u307e\u305f, $\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f$\\mathcal{M}(\\mathbb{R})$\u306b\u5c5e\u3059\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u958b\u533a\u9593$I$\u306b\u5bfe\u3057\u3066$\\Psi$\u304c$I$\u4e0aa.e.\u306b\u591a\u9805\u5f0f\u3068\u4e00\u81f4\u3057\u306a\u3044\u306a\u3089, $\\mathcal{N}_1(\\Psi,W,I)$\u306f$C(\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{thm}\n\\begin{proof}\n  \u958b\u533a\u9593$J = (c,d)$\u3068$\\varepsilon$\u3092$J^{\\varepsilon}:=(c-\\varepsilon,d+\\varepsilon) \\subset I$\u3068\u306a\u308b\u3088\u3046\u306b\u53d6\u308b. $\\Psi$\u306b\u3064\u3044\u3066\u306e\u4eee\u5b9a\u3088\u308a\u4efb\u610f\u306e$\\rho \\in C_0^{\\infty}(\\mathbb{R})$\u306b\u5bfe\u3057\u3066$\\Psi * \\rho$\u306f$C^{\\infty}$\u7d1a\u95a2\u6570\u3067\u3042\u308b(\u4ed8\u9332\u306e\u547d\u984c\\ref{L1locandC0Convolution}\u3092\u53c2\u7167). \u307e\u305f, \u88dc\u984c\\ref{CovolutionUniformAppro}\u3088\u308a, $\\mathrm{supp}(\\rho) \\subset (-\\varepsilon,\\varepsilon)$\u306a\u308b\u4efb\u610f\u306e$\\rho \\in C_0^{\\infty}(\\mathbb{R})$\u3068\u4efb\u610f\u306e$w \\in W, b \\in I$\u306b\u5bfe\u3057\u3066, $x$\u306e\u95a2\u6570\n  \\[\n  (\\Psi * \\rho)(wx + b) = \\int_{\\mathbb{R}} \\Psi(wx+b-y)\\rho(y)dy\n  \\]\n  \u306f$y_i \\in \\mathrm{supp}(\\rho) \\subset (-\\varepsilon,\\varepsilon)$\u3092\u9069\u5f53\u306b\u53d6\u308c\u3070$\\sum_{i} \\Psi(wx+b-y_i)\\rho(y_i)\\Delta y_i$\u3067\u5e83\u7fa9\u4e00\u69d8\u8fd1\u4f3c\u3055\u308c\u308b. \u3086\u3048\u306b, \n  \\[\n  \\overline{\\mathcal{N}_1(\\Psi * \\rho,W,I)} \n  \\subset \\overline{\\mathcal{N}_1(\\Psi,W,J^{\\varepsilon})} \n  \\subset \\overline{\\mathcal{N}_1(\\Psi,W,I)} \n  \\]\n  \u304c\u308f\u304b\u308b(\u305f\u3060\u3057, \u3053\u3053\u3067\u306e\u9589\u5305\u306f\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u3042\u308b). \n  \u3044\u307e$\\Psi * \\rho$\u306f$C^{\\infty}$\u7d1a\u3060\u304b\u3089,  \u88dc\u984c\\ref{NeuralNetApproPoly}\u306e\u8a3c\u660e\u3088\u308a,  \u4efb\u610f\u306e\u81ea\u7136\u6570$k$\u306b\u5bfe\u3057\u3066$x \\mapsto x^k (\\Psi * \\rho)^{(k)}(b)$\u306f$\\overline{\\mathcal{N}_1((\\Psi * \\rho,W,I)}$\u306b\u542b\u307e\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u308b. Stone-Weierstrass\u306e\u5b9a\u7406\u3088\u308a\u591a\u9805\u5f0f\u95a2\u6570\u5168\u4f53\u306f$C(\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u306e\u3067, \u3082\u3057$\\mathcal{N}_1(\\Psi,W,I)$\u304c$C(\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u306a\u3044\u306a\u3089\u3070, \u3042\u308b$m$\u304c\u5b58\u5728\u3057\u3066$\\mathrm{supp}(\\rho) \\subset (-\\varepsilon,\\varepsilon)$\u306a\u308b\u4efb\u610f\u306e$\\rho \\in C_0^{\\infty}(\\mathbb{R})$\u306b\u5bfe\u3057\u3066$(\\Psi * \\rho)^{(m)}(b) = 0 ~(\\forall b \\in I)$\u3068\u306a\u3089\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044. \u3057\u305f\u304c\u3063\u3066, \u3053\u306e\u3068\u304d, $\\mathrm{supp}(\\rho) \\subset (-\\varepsilon,\\varepsilon)$\u306a\u308b\u4efb\u610f\u306e$\\rho \\in C_0^{\\infty}(\\mathbb{R})$\u306b\u5bfe\u3057\u3066$\\Psi * \\rho$\u306f$I$\u4e0a\u3067$m$\u6b21\u4ee5\u4e0b\u306e\u591a\u9805\u5f0f\u3068\u306a\u308b. \u3068\u3053\u308d\u304c, \u88dc\u984c\\ref{ConvolutionAlmostEverywhereConvergence}\u3088\u308a$\\rho_j \\in C_0^{\\infty}(\\mathbb{R})$\u304c\u5b58\u5728\u3057\u3066, $\\mathrm{supp}(\\rho_j) \\subset (-\\varepsilon,\\varepsilon)$\u304b\u3064a.e.$x \\in I$\u306b\u5bfe\u3057\u3066$\\Psi * \\rho_j(x) \\rightarrow \\Psi(x)$\u3068\u306a\u308b. \u3088\u3063\u3066, \u88dc\u984c\\ref{PointwiseConvergencePolyPreserve}\u3088\u308a$\\Psi$\u306f$I$\u4e0a\u3067\u3042\u308b\u591a\u9805\u5f0f\u3068a.e.\u3067\u4e00\u81f4\u3059\u308b. \u3057\u304b\u3057, \u3053\u308c\u306f\u6211\u3005\u306e\u4eee\u5b9a\u306b\u53cd\u3059\u308b. \n\\end{proof}\n\u4ee5\u4e0a\u3067Leshno\u3089\u306e\u4e3b\u7d50\u679c\u304c\u8a3c\u660e\u3067\u304d\u305f. \n\u306a\u304a, \u7b2c2\u7ae0\u3067\u8ff0\u3079\u305f\u3088\u3046\u306bLeshno\u3089\u306e\u7d50\u679c\u306b\u304a\u3051\u308b\u9023\u7d9a\u6027\u306e\u4eee\u5b9a\u306fHornik\u306e\u6307\u6458\\cite{Hornik1993}\u306b\u3088\u308a\u6b21\u306e\u5f62\u306b\u7de9\u548c\u3055\u308c\u305f. \n\\begin{thm*}[Hornik 1993 \\cite{Hornik1993}]\n  $W \\subset \\mathbb{R}^r$\u306f$0$\u3092\u5185\u70b9\u3068\u3057\u3066\u6301\u3064\u3068\u3059\u308b. \n  \u307e\u305f, $\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f\u5404\u9589\u533a\u9593\u4e0a\u3067\u6709\u754c\u304b\u3064Riemann\u7a4d\u5206\u53ef\u80fd\u3067\u3042\u308b\u3068\u3059\u308b. \u3055\u3089\u306b, \u3042\u308b\u958b\u533a\u9593$I$\u306b\u5bfe\u3057\u3066$\\Psi$\u304c$I$\u4e0a\u3067\u591a\u9805\u5f0f\u3068\u4e00\u81f4\u3057\u306a\u3044\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $\\mathcal{A} \\subset \\mathbb{R}^r$\u4e0a\u3067$0$\u3092\u53d6\u308b\u975e\u81ea\u660e\u306a$r$\u5909\u6570\u6589\u6b21\u591a\u9805\u5f0f\u304c\u5b58\u5728\u3057\u306a\u3044\u306a\u3089\u3070, $\\mathcal{N}_r(\\Psi,W,I)$\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{thm*}\n\u3053\u306e\u5b9a\u7406\u306f\u6b21\u306e\u5b9a\u7406\u306b\u542b\u307e\u308c\u308b. \u306a\u305c\u306a\u3089, $0$\u3092\u5185\u70b9\u3068\u3057\u3066\u6301\u3064$W \\subset \\mathbb{R}^r$\u306b\u5bfe\u3057\u3066, $\\mathcal{A} \\subset \\mathbb{R}^r$\u3068\u3057\u3066\u539f\u70b9\u4e2d\u5fc3\u306e\u958b\u7403\u3092\u3068\u308a\u5341\u5206\u5c0f\u3055\u3044$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066$W' = (-\\varepsilon,\\varepsilon)$\u3068\u304a\u3051\u3070$W' \\mathcal{A} \\subset W$\u3068\u306a\u308b\u304b\u3089\u3067\u3042\u308b(\u958b\u7403\u4e0a\u3067\u591a\u9805\u5f0f\u306f\u4e00\u610f\u306b\u6c7a\u5b9a\u3055\u308c\u308b\u3053\u3068\u306b\u6ce8\u610f). \n\\begin{thm}\n  $W \\subset \\mathbb{R}$\u306f\u5b64\u7acb\u70b9\u3092\u6301\u305f\u305a$0 \\in W$\u3067\u3042\u308b\u3068\u3059\u308b. \n  \u307e\u305f, $\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f\u5404\u9589\u533a\u9593\u4e0a\u3067\u6709\u754c\u304b\u3064Riemann\u7a4d\u5206\u53ef\u80fd\u3067\u3042\u308b\u3068\u3059\u308b. \u3055\u3089\u306b, \u3042\u308b\u958b\u533a\u9593$I$\u306b\u5bfe\u3057\u3066$\\Psi$\u304c$I$\u4e0a\u3067\u591a\u9805\u5f0f\u3068\u4e00\u81f4\u3057\u306a\u3044\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $\\mathcal{A} \\subset \\mathbb{R}^r$\u4e0a\u3067$0$\u3092\u53d6\u308b\u975e\u81ea\u660e\u306a$r$\u5909\u6570\u6589\u6b21\u591a\u9805\u5f0f\u304c\u5b58\u5728\u3057\u306a\u3044\u306a\u3089\u3070, \n  \\[\n  \\begin{aligned}\n  \\mathcal{N}_r(\\Psi,W\\mathcal{A},I) = \\mathrm{span}\\{ \\mathbb{R}^r \\ni x \\mapsto \\Psi(w a^{T} x + b) \\in \\mathbb{R} \\mid w \\in W, a \\in \\mathcal{A}, b \\in I \\}\n  \\end{aligned}\n  \\]\n  \u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{thm}\n\u3055\u3089\u306b\u3053\u306e\u5b9a\u7406\u3092\u793a\u3059\u305f\u3081\u306b\u306f, Leshno\u3089\u306e\u7d50\u679c\u306e\u8a3c\u660e\u3068\u540c\u69d8\u306b\u3057\u3066\u6b21\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044. \n\\begin{prop}\n  \u5404\u9589\u533a\u9593\u4e0a\u3067\u6709\u754c\u304b\u3064Riemann\u7a4d\u5206\u53ef\u80fd\u306a$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306b\u5bfe\u3057\u3066, \u4efb\u610f\u306e$\\rho \\in C_0^{\\infty}(\\mathbb{R})$\u306b\u5bfe\u3057\u3066, $\\Psi$\u3068$\\rho$\u306e\u7573\u307f\u8fbc\u307f\u7a4d\n  \\[\n  (\\Psi * \\rho)(x) = \\int_{\\mathbb{R}} \\Psi(x-y) \\rho(y) dy\n  \\]\n  \u306f$\\mathcal{N}_1(\\Psi,\\{1\\},\\mathrm{supp}(\\rho))$\u306b\u3088\u308a\u5e83\u7fa9\u4e00\u69d8\u8fd1\u4f3c\u3055\u308c\u308b\n\\end{prop}\n\\begin{proof}\n  \u7a7a\u3067\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}$\u3092\u4efb\u610f\u306b\u53d6\u308b. $\\alpha> 0$\u3092\u9069\u5f53\u306b\u53d6\u308a$K$\u3068 $\\mathrm{supp}(\\rho)$\u304c$ [-\\alpha,\\alpha]$\u306b\u542b\u307e\u308c\u308b\u3088\u3046\u306b\u3059\u308b. $m \\in \\mathbb{N}$\u306b\u5bfe\u3057\u3066\n  \\[\n  y_j = - \\alpha + 2j\\alpha\/m,~~ \\Delta y_j = 2 \\alpha\/m ~~~~(j=1,\\ldots,m)\n  \\]\n  \u3068\u304a\u304d, \n  \\[\n  x \\mapsto \\sum_{j=1}^{m} \\Psi(x - y_j) \\rho(y_j) \\Delta y_j\n  \\]\n  \u304c$\\Psi * \\rho$\u306b$[-\\alpha,\\alpha]$\u4e0a\u4e00\u69d8\u53ce\u675f\u3059\u308b\u3053\u3068\u3092\u793a\u305d\u3046. \u4efb\u610f\u306e$x \\in [-\\alpha,\\alpha]$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  &\\left\\lvert \\int_{\\mathbb{R}} \\Psi(x-y) \\rho(y) dy - \\sum_{j=1}^{m} \\Psi(x - y_j) \\rho(y_j) \\Delta y_j \\right\\rvert \\\\\n  &\\leq \\left| \\int_{\\mathbb{R}} \\Psi(x-y) \\rho(y) dy - \\sum_{j=1}^m \\int_{\\Delta_j} \\Psi(x - y_j) \\rho(y) dy \\right| \\\\\n  &~~~~ + \\left| \\sum_{j=1}^m \\int_{\\Delta_j} \\Psi(x - y_j) \\rho(y) dy - \\sum_{j=1}^m \\Psi(x-y_j)\\rho(y_j)\\Delta y_j \\right|\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u53f3\u8fba\u306e\u4e8c\u3064\u306e\u9805\u304c\u305d\u308c\u305e\u308c$[-\\alpha,\\alpha]$\u4e0a\u4e00\u69d8\u306b$0$\u306b\u53ce\u675f\u3059\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044. \n  \u307e\u305a, \u7b2c\u4e00\u9805\u306b\u3064\u3044\u3066\u8003\u3048\u308b. $\\mathrm{supp}(\\rho) \\subset [-\\alpha,\\alpha]$\u3067\u3042\u308b\u304b\u3089, \u4efb\u610f\u306e$x \\in [-\\alpha,\\alpha]$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  &\\left| \\int_{\\mathbb{R}} \\Psi(x-y) \\rho(y) dy - \\sum_{j=1}^m \\int_{\\Delta_j} \\Psi(x - y_j) \\rho(y) dy \\right| \\\\\n  &\\leq \\sum_{j=1}^m \\int_{\\Delta_j} |\\Psi(x-y) - \\Psi(x-y_j)||\\rho(y)| dy \\\\\n  &\\leq \\lVert \\rho \\rVert_{L^{\\infty}} \\sum_{j=1}^m \\left(  \\sup_{y \\in \\Delta_j} \\Psi(x-y) - \\inf_{y \\in \\Delta_j} \\Psi(x-y) \\right)\\frac{2\\alpha}{m}\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. $[-2\\alpha,2\\alpha]$\u3092$2m$\u7b49\u5206\u3059\u308b\u5206\u5272$\\Delta'_j~(j=1,\\ldots,2m)$\u306b\u3064\u3044\u3066, \n  \\[\n  \\begin{aligned}\n  &\\sum_{j=1}^m \\left( \\sup_{y \\in \\Delta_j} \\Psi(x-y) - \\inf_{y \\in \\Delta_j} \\Psi(x-y) \\right) \\\\\n  &\\leq \\sum_{j=1}^{2m} \\left( \\sup_{z \\in \\Delta'_j} \\Psi(z) - \\inf_{z \\in \\Delta'_j} \\Psi(z) \\right) \n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b\u306e\u3067, $\\Psi$\u304c$[-2\\alpha,2\\alpha]$\u4e0aRiemann\u7a4d\u5206\u53ef\u80fd\u3067\u3042\u308b\u3053\u3068\u3088\u308a, $[-\\alpha,\\alpha]$\u4e0a\u4e00\u69d8\u306b\n  \\[\n  \\begin{aligned}\n  &\\left| \\int_{\\mathbb{R}} \\Psi(x-y) \\rho(y) dy - \\sum_{j=1}^m \\int_{\\Delta_j} \\Psi(x - y_j) \\rho(y) dy \\right| \\\\\n  &\\leq \\lVert \\rho \\rVert_{L^{\\infty}} \\sum_{j=1}^{2m} \\left( \\sup_{z \\in \\Delta'_j} \\Psi(z) - \\inf_{z \\in \\Delta'_j} \\Psi(z) \\right) \\frac{2\\alpha}{m} \\\\\n  &\\rightarrow \\lVert \\rho \\rVert_{L^{\\infty}} \\left( \\int_{-2\\alpha}^{2\\alpha} \\Psi(z) dz - \\int_{-2\\alpha}^{2\\alpha} \\Psi(z) dz \\right) = 0 ~~(m \\rightarrow \\infty)\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \n  \u6b21\u306b\u7b2c\u4e8c\u9805\u306b\u3064\u3044\u3066\u8003\u3048\u3088\u3046. $\\varepsilon > 0$\u3092\u4efb\u610f\u306b\u53d6\u308b. \u4efb\u610f\u306e$x \\in [-\\alpha,\\alpha]$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  &\\left| \\sum_{j=1}^m \\int_{\\Delta_j} \\Psi(x - y_j) \\rho(y) dy - \\sum_{j=1}^m \\Psi(x-y_j)\\rho(y_j)\\Delta y_j \\right| \\\\\n  &= \\left| \\sum_{j=1}^m \\int_{\\Delta_j} \\Psi(x - y_j) ( \\rho(y)  - \\rho(y_j))dy \\right| \\\\\n  &\\leq \\sum_{j=1}^m  \\int_{\\Delta_j} |\\Psi(x - y_j)|~| \\rho(y)  - \\rho(y_j)| dy \\\\\n  &\\leq \\lVert \\Psi \\rVert_{L^{\\infty}([-2\\alpha,2\\alpha])} \\sum_{j=1}^m  \\int_{\\Delta_j} | \\rho(y)  - \\rho(y_j)| dy \n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u305d\u3053\u3067, $\\rho$\u304c$\\mathbb{R}$\u4e0a\u4e00\u69d8\u9023\u7d9a\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066, $N \\in \\mathbb{N}$\u3092\u6b21\u3092\u307f\u305f\u3059\u3088\u3046\u306b\u53d6\u308b.  \n  \\[\n    |s-t| < 2\\alpha\/N \\Rightarrow |\\rho(s) - \\rho(t)| \\leq \\frac{\\varepsilon}{2\\alpha \\lVert \\Psi \\rVert_{L^{\\infty}([-2\\alpha,2\\alpha])}}. \n  \\]\n  \u3059\u308b\u3068, \u4efb\u610f\u306e$m \\geq N$\u3068$x \\in [-\\alpha,\\alpha]$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n    &\\left| \\sum_{j=1}^m \\int_{\\Delta_j} \\Psi(x - y_j) \\rho(y) dy - \\sum_{j=1}^m \\Psi(x-y_j)\\rho(y_j)\\Delta y_j \\right| \\\\\n    &\\leq \\lVert \\Psi \\rVert_{L^{\\infty}([-2\\alpha,2\\alpha])} \\sum_{j=1}^m  \\int_{\\Delta_j} | \\rho(y)  - \\rho(y_j)| dy \\leq \\varepsilon\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u4ee5\u4e0a\u3067\u8a3c\u660e\u3067\u304d\u305f. \n\\end{proof}\n\\begin{cor}\n  $\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f\u5404\u9589\u533a\u9593\u4e0a\u3067\u6709\u754c\u304b\u3064Riemann\u7a4d\u5206\u53ef\u80fd\u3067\u3042\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $\\Sigma^r(\\Psi)$\u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3068, $\\Psi$\u304c\u3044\u304b\u306a\u308b\u591a\u9805\u5f0f\u95a2\u6570\u3068\u3082a.e.\u3067\u4e00\u81f4\u3059\u308b\u3053\u3068\u306f\u306a\u3044\u3053\u3068\u306f\u540c\u5024\u3067\u3042\u308b. \n\\end{cor}\n\n\\subsection{Chui\u3068Li\u306e\u7d50\u679c}\n\u672c\u7bc0\u3067\u306fCybenko\u306e\u7d50\u679c\\cite{Cybenko}\u3092\u542b\u3080Chui\u3068Li\u306b\u3088\u308b\u6b21\u306e\u7d50\u679c\u3092\u8a3c\u660e\u3059\u308b. \n\\begin{thm}[Chui and Li 1992 \\cite{ChuiAndLi}]\\label{UATByChuiandLi} \\\\\n$\\sigma : \\mathbb{R} \\rightarrow \\mathbb{R}$ \u3092\u9023\u7d9a\u306asigmoidal\u95a2\u6570\u3068\u3059\u308b\u3068,\n\\[\n\\mathcal{N}_r(\\sigma,\\mathbb{Z}^r,\\mathbb{Z}) = \\mathrm{span} \\{\\mathbb{R}^r \\ni x \\mapsto \\sigma(m^{\\mathrm{T}} x + k) \\in \\mathbb{R} \\mid m \\in \\mathbb{Z}^r, k \\in \\mathbb{Z} \\}\n\\]\n\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{thm}\n\u3053\u306e\u5b9a\u7406\u3092\u3075\u305f\u3064\u306e\u88dc\u984c\u306b\u5206\u3051\u3066\u8a3c\u660e\u3059\u308b. \u88dc\u984c\u306e\u4e3b\u5f35\u3092\u8ff0\u3079\u308b\u305f\u3081\u306b\u8a00\u8449\u3068\u8a18\u53f7\u306e\u5b9a\u7fa9\u3092\u3057\u3066\u304a\u304f.\n\n$X \\subset \\mathbb{R}^r$\u306b\u5bfe\u3057\u3066, \u4f4d\u76f8\u7a7a\u9593$(X, \\mathcal{O})$\u3092$\\mathbb{R}^r$\u306e\u90e8\u5206\u7a7a\u9593\u3068\u3059\u308b.  \u3059\u306a\u308f\u3061$\\mathcal{O} = \\{X \\cap U \\mid U \\subset \\mathbb{R}^r, \\mathrm{open} \\}$\u3068\u3059\u308b. \u307e\u305f, $(X, \\mathcal{O})$\u4e0a\u306e\u6b63\u5247\u7b26\u53f7\u4ed8\u304dBorel\u6e2c\u5ea6\u5168\u4f53\u306e\u96c6\u5408\u3092$M(X)$\u3067\u8868\u3059\u3053\u3068\u306b\u3059\u308b(\u6b63\u5247\u7b26\u53f7\u4ed8\u304d\u6e2c\u5ea6\u306e\u5b9a\u7fa9\u306b\u3064\u3044\u3066\u306f\u4ed8\u9332\u3092\u53c2\u7167\u3055\u308c\u305f\u3044). \n\n\\begin{defn}[discriminatory\u95a2\u6570] \\\\\n\u95a2\u6570 $\\sigma : \\mathbb{R} \\rightarrow \\mathbb{R}$ \u304c\u6b21\u306e\u6761\u4ef6\u3092\u307f\u305f\u3059\u3068\u304d, discriminatory\u95a2\u6570\u3068\u547c\u3076: \u4efb\u610f\u306e\u7a7a\u3067\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$X \\subset \\mathbb{R}^r$\u306b\u5bfe\u3057\u3066, \n\\[\n\\forall \\mu \\in M(X), \\left(\\forall m \\in \\mathbb{Z}^r, k \\in \\mathbb{Z}, \\int_{X}{\\sigma \\left(m^{\\mathrm{T}} x + k \\right)}{d\\mu(x)} = 0 \\right) \\Rightarrow \\mu = 0. \n\\]\n\\end{defn}\n\n\u3053\u306e\u5b9a\u7fa9\u306e\u3082\u3068\u3067\u4ee5\u4e0b\u306e\u3075\u305f\u3064\u306e\u547d\u984c\u304c\u6210\u308a\u7acb\u3064. \n\n\\begin{lem}\\label{UATofDiscriminatoryFunc} \\\\\n$\\sigma : \\mathbb{R} \\rightarrow \\mathbb{R}$ \u3092\u9023\u7d9a\u306adiscriminatory\u95a2\u6570\u3068\u3059\u308b\u3068, $\\mathcal{N}_r(\\sigma,\\mathbb{Z}^r,\\mathbb{Z})$\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{lem}\n\n\\begin{lem}\\label{discriminatory} \\\\\n\u6709\u754c\u3067\u53ef\u6e2c\u306asigmoidal\u95a2\u6570$\\sigma$\u306fdiscriminatory\u95a2\u6570\u3067\u3042\u308b. \u7279\u306b\u9023\u7d9a\u306asigmoidal\u95a2\u6570\u306fdiscriminatory\u95a2\u6570\u3067\u3042\u308b. \n\\end{lem}\n\n\u4e0a\u8a18\u3075\u305f\u3064\u306e\u88dc\u984c\u304b\u3089\u5b9a\u7406\\ref{UATByChuiandLi}\u304c\u5c0e\u304b\u308c\u308b\u3053\u3068\u306f\u660e\u3089\u304b\u3067\u3042\u308d\u3046. \u4ee5\u4e0b\u3067\u3053\u308c\u3089\u306e\u88dc\u984c\u306e\u8a3c\u660e\u3092\u3057\u3066\u3044\u304f. \n\n\\renewcommand{\\proofname}{\\bf{\u88dc\u984c}\\ref{UATofDiscriminatoryFunc}\\bf{\u306e\u8a3c\u660e}}\n\n\\begin{proof} \\\\\n$K \\subset \\mathbb{R}^r$\u3092\u7a7a\u3067\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u3068\u3059\u308b.  $\\mathcal{N}_r(\\sigma,\\mathbb{Z}^r,\\mathbb{Z})$\u306e\u5143(\u95a2\u6570)\u306e$K$\u4e0a\u3078\u306e\u5236\u9650\u3059\u3079\u3066\u3092\u96c6\u3081\u305f\u96c6\u5408\u3092$S$\u3068\u304a\u304f. $S$\u306f $C(K)$ \u306e\u90e8\u5206\u7dda\u5f62\u7a7a\u9593\u3067\u3042\u308b. \u88dc\u984c\\ref{UATofDiscriminatoryFunc}\u306e\u4e3b\u5f35\u306f$\\overline{S} =  C(K)$\u304c\u6210\u308a\u7acb\u3064\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b. \u305d\u3053\u3067, $\\overline{S} \\neq C(K)$\u3092\u4eee\u5b9a\u3059\u308b(\u80cc\u7406\u6cd5). \u3059\u308b\u3068,  Hahn-Banach\u306e\u62e1\u5f35\u5b9a\u7406\u306b\u3088\u308a, $C(K)$\u4e0a\u306e\u6709\u754c\u7dda\u5f62\u6c4e\u95a2\u6570$L \\neq 0$\u3067, $L = 0 ~(\\mathrm{on} ~\\overline{S})$ \u306a\u308b\u3082\u306e\u304c\u53d6\u308c\u308b(\u4ed8\u9332\u306e\u7cfb\\ref{HahnBanachCor1}\u53c2\u7167). $K$\u306f\u30b3\u30f3\u30d1\u30af\u30c8Hausdorff\u7a7a\u9593\u3067\u3042\u308b\u306e\u3067, Riesz-Markov-\u89d2\u8c37\u306e\u8868\u73fe\u5b9a\u7406\u304b\u3089, $\\mu \\in M(K)$\u3067, \\[\n\\forall f \\in C(K), \\\\ L(f) = \\int_{K}{f}{d\\mu}\n\\]\n\u3092\u307f\u305f\u3059\u3082\u306e\u304c\u53d6\u308c\u308b. \u4efb\u610f\u306e$m \\in \\mathbb{Z}^n, k \\in \\mathbb{Z}$\u306b\u5bfe\u3057\u3066, \u5199\u50cf\n\\[\nK \\ni x \\mapsto \\sigma(m^{\\mathrm{T}} x + k) \\in \\mathbb{R}\n\\]\n\u306f$S$\u306b\u5c5e\u3059\u308b\u306e\u3067, \n\\[\n\\int_{K} {\\sigma(m^{\\mathrm{T}} x + k)}{d\\mu(x)} = 0\n\\]\n\u3068\u306a\u308b. \u3086\u3048\u306b, $\\sigma$\u304cdiscriminatory\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u304b\u3089$\\mu = 0$\u3068\u306a\u308a, \u3057\u305f\u304c\u3063\u3066$L = 0$\u3068\u306a\u308b\u304c, \u3053\u308c\u306f$L \\neq 0$\u306b\u53cd\u3059\u308b. \u3088\u3063\u3066, $\\overline{S} = C(K)$\u3067\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044. \n\\end{proof}\n\n\\renewcommand{\\proofname}{\\bf{\u88dc\u984c}\\ref{discriminatory}\\bf{\u306e\u8a3c\u660e}}\n\n\\begin{proof} \\\\\n\u5f8c\u534a\u306b\u3064\u3044\u3066\u306f\u9023\u7d9a\u6027\u304b\u3089\u53ef\u6e2c\u6027\u304c\u308f\u304b\u308a, sigmoidal\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u3068\u9023\u7d9a\u6027\u304b\u3089\u6709\u754c\u6027\u304c\u308f\u304b\u308b\u306e\u3067\u6210\u7acb\u3059\u308b. \u524d\u534a\u3092\u8a3c\u660e\u3059\u308b. $X \\subset \\mathbb{R}^r$\u3092\u7a7a\u3067\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u3068\u3059\u308b. \n$\\mu \\in M(X)$\u3092\u4efb\u610f\u306b\u3068\u308a\u56fa\u5b9a\u3059\u308b. \u305d\u3057\u3066, \n\\[\n\\forall m \\in \\mathbb{Z}^r, k \\in \\mathbb{Z}, ~\\int_{X}{\\sigma \\left(m^{\\mathrm{T}} x + k \\right)}{d\\mu(x)} = 0 \n\\]\n\u304c\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3059\u308b. \n\u3053\u306e\u3068\u304d, $\\mu = 0$\u3092\u793a\u3059\u306e\u304c\u76ee\u6a19\u3067\u3042\u308b. \n\u3055\u3066, \u3044\u307e, \u4efb\u610f\u306e$m \\in \\mathbb{Z}^r, q \\in \\mathbb{Q}, k \\in \\mathbb{Z}$\u306b\u5bfe\u3057\u3066, \n\\[\n\\Pi_{m, q} = \\left\\{ x \\in X \\mid m^{\\mathrm{T}} x + q = 0 \\right\\} , ~~\nH_{m, q} = \\left \\{ x \\in X \\mid m^{\\mathrm{T}} x + q > 0 \\right \\}\n\\]\n\u3068\u304a\u304d, \n\\[\n\\gamma(x) :=\n\\begin{cases}\n1 & x \\in H_{m, q} \\\\\n\\sigma(k) & x \\in \\Pi_{m, q} \\\\\n0 & \\mathrm{otherwise}\n\\end{cases}\n\\]\n\u3068\u5b9a\u3081\u308b\u3068,  $\\sigma$\u306fsigmoidal\u95a2\u6570\u306a\u306e\u3067, $+\\infty$\u306b\u767a\u6563\u3059\u308b\u81ea\u7136\u6570\u5217$l_n$\u3092\u4efb\u610f\u306b\u53d6\u308b\u3068, \u4efb\u610f\u306e$x \\in \\mathbb{R}^r$\u306b\u5bfe\u3057\u3066, \n\\[\n\\lim_{n \\rightarrow \\infty} \\sigma(l_n (m^{\\mathrm{T}} x + q) + k) \\rightarrow \\gamma(x)\n\\]\n\u3068\u306a\u308b. \u305d\u3053\u3067, $q = q_1\/q_2 , ~q_1 \\in \\mathbb{Z}, q_2 \\in \\mathbb{N}$\u3068\u8868\u3059\u3053\u3068\u306b\u3059\u308b\u3068, \n$\\sigma$\u306e\u6709\u754c\u6027\u3068$\\mu$\u306e\u6709\u9650\u6027\u304a\u3088\u3073\u4eee\u5b9a\u3088\u308a, \n\\[\n\\begin{aligned}\n0 &= \\lim_{n \\rightarrow +\\infty}{\\int_{X}{\\sigma(n q_2 (m^{\\mathrm{T}} x + q) + k)}{d\\mu(x)}} \\\\\n&= \\int_{X}{\\lim_{n \\rightarrow +\\infty}\\sigma(n q_2 (m^{\\mathrm{T}} x  + q) + k)}{d\\mu(x)} \\\\\n&= \\int_{X}{\\gamma(x)}{d\\mu(x)}\\\\\n&= \\sigma(k)\\mu\\left( \\Pi_{m, q} \\right)+\\mu\\left( H_{m, q} \\right)\n\\end{aligned} \n\\]\n\u3068\u306a\u308b(2\u3064\u76ee\u306e\u7b49\u53f7\u306f\u512a\u53ce\u675f\u5b9a\u7406\u306b\u3088\u308b). \u3088\u3063\u3066, $\\sigma$\u304csigmoidal\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u3068$k \\in \\mathbb{Z}$\u304c\u4efb\u610f\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u3068, $k \\rightarrow -\\infty$\u3068\u3059\u308b\u3053\u3068\u3067, $\\mu\\left( H_{m, q} \\right) = 0$\u304c\u5f97\u3089\u308c, \u3055\u3089\u306b$k \\rightarrow +\\infty$\u3068\u3059\u308b\u3053\u3068\u3067$\\mu\\left( \\Pi_{m, q} \\right) = 0$\u304c\u5f97\u3089\u308c\u308b. \u3057\u305f\u304c\u3063\u3066,  $\\mathbb{R}$\u4e0a\u306e\u6709\u754c\u306a\u53ef\u6e2c\u95a2\u6570\u306e\u306a\u3059\u7dda\u5f62\u7a7a\u9593\u4e0a\u306e\u7dda\u5f62\u6c4e\u95a2\u6570$F$\u3092\n\\[\nF(h) = \\int_{X}{h(m^{\\mathrm{T}} x)}{d\\mu(x)}\n\\]\n\u306b\u3088\u308a\u5b9a\u3081\u308b\u3068, \u5b9a\u7fa9\u95a2\u6570\n\\[\nh(x) = \\chi_{[q, \\infty)}(x) := \n\\begin{cases}\n1 & (x \\in [q, \\infty)) \\\\\n0 & (x \\notin [q, \\infty))\n\\end{cases}\n\\]\n\u306b\u5bfe\u3057\u3066, \n\\[\nF(h) = \\int_{X}{h(m^{\\mathrm{T}} x)}{d\\mu(x)} = \\mu\\left( \\Pi_{m, -q} \\right)+\\mu\\left( H_{m, -q} \\right) = 0\n\\]\n\u3068\u306a\u308b. \u3057\u305f\u304c\u3063\u3066, $q \\in \\mathbb{Q}$\u306f\u4efb\u610f\u3067\u3042\u3063\u305f\u304b\u3089,  $F$\u306e\u7dda\u5f62\u6027\u3088\u308a\u6709\u7406\u6570\u3092\u7aef\u70b9\u3068\u3059\u308b\u4efb\u610f\u306e$\\mathbb{R}$\u306e\u533a\u9593$J$\u306b\u3064\u3044\u3066$F(\\chi_{J}) = 0$\u3068\u306a\u308b.  \u3057\u305f\u304c\u3063\u3066, \u6709\u9650\u500b\u306e$\\mathbb{R}$\u306e\u533a\u9593\u306e\u5b9a\u7fa9\u95a2\u6570\u306e\u7dda\u5f62\u7d50\u5408\u3092$F$\u3067\u5199\u3059\u3068$0$\u306b\u306a\u308b. \u4ee5\u4e0b, \u4e00\u65e6$m \\in \\mathbb{Z}^r$\u3092\u56fa\u5b9a\u3057\u3066\u8003\u3048\u308b. \u3053\u306e\u3068\u304d\n$A = \\{m^{\\mathrm{T}} x \\mid x \\in X \\}$\u306f$\\mathbb{R}$\u306e\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u3067\u3042\u308b. \u4f55\u6545\u306a\u3089$A$\u306f\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$X$\u306e\u9023\u7d9a\u5199\u50cf\u306b\u3088\u308b\u50cf\u3060\u304b\u3089\u3067\u3042\u308b. \u305d\u3053\u3067\u6709\u7406\u6570\u3092\u7aef\u70b9\u3068\u3059\u308b\u6709\u754c\u9589\u533a\u9593$J$\u3092$A \\subset J$\u3092\u307f\u305f\u3059\u3088\u3046\u306b\u53d6\u308c\u308b.  \u3044\u307e$J$\u4e0a\u306e\u4efb\u610f\u306e\u9023\u7d9a\u95a2\u6570$g$\u306f\u6709\u7406\u6570\u3092\u7aef\u70b9\u3068\u3059\u308b$\\mathbb{R}$\u306e\u533a\u9593\u306e\u5b9a\u7fa9\u95a2\u6570\u305f\u3061\u306e\u7dda\u5f62\u7d50\u5408$\\sum_{k=1}^{N}{a_k \\chi_{J_k}}$\u306b\u3088\u308a\u4e00\u69d8\u8fd1\u4f3c\u3067\u304d\u308b.  \u3059\u306a\u308f\u3061\u4efb\u610f\u306e\u6b63\u6570$\\varepsilon$\u306b\u5bfe\u3057\u3066, \u9069\u5f53\u306a$a_k \\in \\mathbb{R}$\u3068\u6709\u7406\u6570\u3092\u7aef\u70b9\u3068\u3059\u308b\u533a\u9593$J_k$\u3092 \u53d6\u308c\u3070, \n\\[\n\\forall x \\in J,  ~ \\left\\lvert g(x) - \\sum_{k=1}^{N}{a_k \\chi_{J_k}(x)} \\right\\rvert < \\varepsilon\n\\]\n\u3068\u3067\u304d\u308b. \u5b9f\u969b,  \u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u4e0a\u306e\u9023\u7d9a\u95a2\u6570\u306f\u4e00\u69d8\u9023\u7d9a\u306a\u306e\u3067, \u4efb\u610f\u306e$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, $\\delta>0$\u3092\u9069\u5f53\u306b\u53d6\u308c\u3070, \n\\[\n\\forall x, y \\in J,  |x-y|<\\delta \\Rightarrow |g(x)-g(y)|<\\varepsilon\n\\]\n\u304c\u6210\u308a\u7acb\u3064. \u305d\u3053\u3067, \u4e00\u3064\u306e\u533a\u9593\u306e\u5e45\u304c$\\delta$\u306b\u53ce\u307e\u308b\u3088\u3046\u306a\u5206\u5272$J = \\sum_{k=1}^{N}{J_k}$\u3067\u3042\u3063\u3066, \u5404\u533a\u9593$J_k$\u306e\u7aef\u70b9\u304c\u6709\u7406\u6570\u306b\u306a\u308b\u3082\u306e\u3092\u53d6\u308b. \u305d\u3057\u3066\u5404$k$\u306b\u5bfe\u3057\u3066$x_k \\in J_k$\u3092\u4efb\u610f\u306b\u53d6\u308c\u3070, \n\\[\n\\forall x \\in J,~~ \\left\\lvert g(x) - \\sum_{k=1}^{N}{g(x_k) \\chi_{J_k}(x)} \\right\\rvert < \\varepsilon\n\\]\n\u3068\u306a\u308b. \u3057\u305f\u304c\u3063\u3066, \n\\[\n\\begin{aligned}\n\\left\\lvert F(g)\\right\\rvert \n&= \\left \\lvert \\int_{X}{g(m^{\\mathrm{T}} x)}{d\\mu(x)} \\right \\rvert \\\\\n&= \\left \\lvert \\int_{X}{g(m^{\\mathrm{T}} x)}{d\\mu(x)} - \\int_{X}{\\sum_{k=1}^{N}{a_k \\chi_{J_k}(m^{\\mathrm{T}} x)}}{d\\mu(x)} \\right \\rvert \\\\\n&\\leq \\int_{X}{\\left \\lvert g(m^{\\mathrm{T}} x) - \\sum_{k=1}^{N}{a_k \\chi_{J_k}(m^{\\mathrm{T}} x)}\\right \\lvert}{d\\left\\rvert\\mu\\right\\rvert(x)} \\\\\n&\\leq \\int_{X}{\\varepsilon}{d\\lvert\\mu\\rvert(x)} = \\varepsilon \\left\\lvert \\mu \\right\\rvert(X)\n\\end{aligned} \n\\]\n\u3068\u306a\u308b. \u305f\u3060\u3057$\\left\\lvert\\mu\\right\\rvert = \\mu^{+}+\\mu^{-}$\u3067\u3042\u308b. \u3088\u3063\u3066, $\\left\\lvert\\mu\\right\\rvert(X)$\u304c$\\varepsilon$\u306b\u4f9d\u5b58\u3057\u306a\u3044\u5b9a\u6570(\u6709\u9650\u5024)\u3067\u3042\u308b\u3053\u3068\u3068$\\varepsilon$\u306e\u4efb\u610f\u6027\u304b\u3089, $F(g) = 0$\u304c\u308f\u304b\u308b. \u7279\u306b$\\cos(x), \\sin(x)$\u306b\u3064\u3044\u3066\u3082\u3053\u308c\u304c\u6210\u7acb\u3059\u308b\u306e\u3067, \n\\[\n0 = \\int_{X}{\\cos(m^{\\mathrm{T}} x)+i\\sin(m^{\\mathrm{T}} x)}{d\\mu(x)} = \\int_{X}{\\exp(im^{\\mathrm{T}} x)}{d\\mu(x)}\n\\]\n\u304c\u308f\u304b\u308b. \u3088\u3063\u3066, $m \\in \\mathbb{Z}$\u306f\u4efb\u610f\u3060\u3063\u305f\u306e\u3067Stone-Weierstrass\u306e\u5b9a\u7406(\u4ed8\u9332\u306e\u7cfb\\ref{CorforChuiandLi})\u304b\u3089$\\mu = 0$\u3068\u306a\u308b. \u3053\u308c\u304c\u793a\u3057\u305f\u3044\u3053\u3068\u3067\u3042\u3063\u305f. \n\\end{proof}\n\n\\renewcommand{\\proofname}{\\bf{\u8a3c\u660e}}\n\n\\subsection{Sun\u3068Cheney \u306e\u7d50\u679c}\n\u672c\u7bc0\u3067\u306f Chui\u3068Li \u306e\u7d50\u679c\u3092\u81ea\u7136\u306b\u30ce\u30eb\u30e0\u7a7a\u9593$X$\u4e0a\u306e\u9023\u7d9a\u95a2\u6570\u7a7a\u9593$C(X)$\u306b\u62e1\u5f35\u3057\u305f Sun\u3068Cheney \u306b\u3088\u308b\u6b21\u306e\u7d50\u679c\u3092\u8a3c\u660e\u3059\u308b. \n\u307e\u305f, mean-periodic\u95a2\u6570\u306a\u3069\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406\u306b\u95a2\u9023\u3059\u308b\u4e8b\u67c4\u3092\u3044\u304f\u3064\u304b\u7d39\u4ecb\u3059\u308b. \n\\begin{thm}[Sun and Cheney 1992 \\cite{SunAndCheney}] \\\\\n  $X \\neq \\{0\\}$\u3092\u30ce\u30eb\u30e0\u7a7a\u9593\u3068\u3057, $\\mathcal{F} \\subset X^*$\u306f\n  \\[\n  \\{ f\/\\lVert f \\rVert \\mid f \\in \\mathcal{F}, f \\neq 0 \\}\n  \\]\n  \u304c$X^*$\u306e\u5358\u4f4d\u7403\u9762$\\{f \\in X^* \\mid \\lVert f \\rVert = 1\\}$\u306b\u304a\u3044\u3066\u4f5c\u7528\u7d20\u30ce\u30eb\u30e0$\\lVert \\cdot \\rVert$\u306b\u95a2\u3057\u3066\u7a20\u5bc6\u3067\u3042\u308b\u3082\u306e\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u9023\u7d9a\u306asigmoidal\u95a2\u6570$\\sigma:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\mathcal{G} = \\{\\mathbb{R} \\ni t \\mapsto \\sigma(kt+j) \\in \\mathbb{R} \\mid k,j \\in \\mathbb{Z} \\}\n  \\]\n  \u3068\u304a\u304f\u3068, \n  \\[\n  \\mathcal{G} \\circ \\mathcal{F} = \\{X \\ni x \\mapsto \\sigma(kf(x)+j) \\in \\mathbb{R} \\mid k,j \\in \\mathbb{Z}, f \\in \\mathcal{F} \\}\n  \\]\n  \u306e\u7dda\u5f62\u5305\u306f$C(X)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{thm}\n\u3053\u306e\u5b9a\u7406\u3092\u793a\u3059\u305f\u3081\u306b\u3072\u3068\u3064\u5b9a\u7406\u3092\u793a\u3059.  \u306a\u304a\u672c\u7bc0\u3067\u306f\u8a18\u8ff0\u3092\u7c21\u6f54\u306b\u3059\u308b\u305f\u3081\u306b\u6b21\u306e\u8a00\u8449\u3092\u7528\u3044\u308b. \n\\begin{defn}\n  $X$\u3092\u30ce\u30eb\u30e0\u7a7a\u9593\u3068\u3059\u308b\u3068\u304d, \u90e8\u5206\u96c6\u5408$S \\subset C(X)$\u304c\u57fa\u790e\u7684\u3067\u3042\u308b\u3068\u306f, $S$\u306e\u7dda\u5f62\u5305\u304c\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067$C(X)$\u306b\u304a\u3044\u3066\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3092\u3044\u3046. \n\\end{defn}\n\\begin{thm}\\label{MainResult2}\n  $X \\neq \\{0\\}$\u3092\u30ce\u30eb\u30e0\u7a7a\u9593\u3068\u3057, $\\mathcal{G} \\subset C(\\mathbb{R})$\u306f\u57fa\u790e\u7684\u3067\u3042\u308b\u3068\u3059\u308b.  \u3053\u306e\u3068\u304d, $\\mathcal{F} \\subset X^*$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\{ f\/\\lVert f \\rVert \\mid f \\in \\mathcal{F}, f \\neq 0 \\}\n  \\]\n  \u304c$X^*$\u306e\u5358\u4f4d\u7403\u9762$\\{f \\in X^* \\mid \\lVert f \\rVert = 1\\}$\u306b\u304a\u3044\u3066\u4f5c\u7528\u7d20\u30ce\u30eb\u30e0$\\lVert \\cdot \\rVert$\u306b\u95a2\u3057\u3066\u7a20\u5bc6\u3067\u3042\u308b\u306a\u3089\u3070, $\\mathcal{G} \\circ \\mathcal{F} = \\{ g \\circ f \\mid g \\in \\mathcal{G}, f \\in \\mathcal{F} \\}$\u306f$C(X)$\u306b\u304a\u3044\u3066\u57fa\u790e\u7684\u3067\u3042\u308b. \n\\end{thm}\n\u3053\u308c\u304c\u793a\u3055\u308c\u308c\u3070, Chui\u3068Li\u306e\u7d50\u679c\u3088\u308a$\\mathcal{G}$\u306f$C(\\mathbb{R})$\u306b\u304a\u3044\u3066\u57fa\u790e\u7684\u3067\u3042\u308b\u306e\u3067\u672c\u7bc0\u306e\u30e1\u30a4\u30f3\u306e\u5b9a\u7406\u306e\u6210\u7acb\u306f\u660e\u3089\u304b\u3067\u3042\u308b. \n\\begin{proof}\n  $h \\in C(X)$\u3068\u3057, \u4efb\u610f\u306b\u7a7a\u3067\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset X$\u3068$\\varepsilon > 0$\u3092\u3068\u308b. \u3059\u308b\u3068, \u5b9a\u7406\\ref{densityRidgeFunc}\u3088\u308a,  $h_i \\in C(\\mathbb{R})$\u3068$\\psi_i \\in X^*$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\sup_{x \\in K} \\left|h(x) - \\sum_{i=0}^n h_i(\\psi_i(x))\\right| < \\varepsilon\/3\n  \\]\n  \u3068\u306a\u308b. $h_i$\u305f\u3061\u3092\u5b9a\u6570\u500d\u3059\u308b\u3053\u3068\u3067$\\lVert \\psi_i \\rVert = 1$\u3068\u4eee\u5b9a\u3057\u3066\u3088\u3044($0$\u304c\u3042\u308b\u5834\u5408\u306f$\\mathbb{R} \\ni t \\mapsto h_i(0) \\in \\mathbb{R}$\u3068\u3044\u3046\u5b9a\u6570\u95a2\u6570\u3092\u5c0e\u5165\u3059\u308c\u3070\u3088\u3044). \u3053\u3053\u3067, $M = \\sup_{x \\in K} \\lVert x \\rVert$\u3068\u304a\u304d, $\\delta > 0$\u3092\u6b21\u3092\u307f\u305f\u3059\u3088\u3046\u306b\u53d6\u308b($h_i$\u306f\u9023\u7d9a\u306a\u306e\u3067\u53d6\u308c\u308b). \n  \\[\n  |h_i(s) - h_i(t)| < \\varepsilon\/3n ~~(|s|\\leq M , |t| \\leq M, |s-t|< M \\delta, 1 \\leq i \\leq n).\n  \\]\n  \u305d\u3057\u3066, $f_i \\in \\mathcal{F}$\u3092$\\lVert f_i \/ \\lVert f_i \\rVert - \\psi_i \\rVert < \\delta$\u3092\u307f\u305f\u3059\u3088\u3046\u306b\u9078\u3076. \n  \u307e\u305f, \u5404$i$\u306b\u3064\u3044\u3066$\\lambda_i = 1\/ \\lVert f_i \\rVert$\u3068\u304a\u304d, $a_{i,j} \\in \\mathbb{R}$\u3068$g_{i,j} \\in \\mathcal{G}$\u3092\n  \\[\n  \\sup_{|t| \\leq M \/ \\lambda_i} \\left|h_i(\\lambda_i t) - \\sum_{j=0}^{N_i} a_{i,j} g_{i,j}(t)\\right| < \\varepsilon\/3n \n  \\]\n  \u3068\u306a\u308b\u3088\u3046\u306b\u3068\u308b. \u3059\u308b\u3068, \u4efb\u610f\u306e$x \\in K$\u306b\u5bfe\u3057\u3066, \n  \\[\n  |f_i(x)| \\leq \\lVert f_i \\rVert \\lVert x \\rVert \\leq M \\lVert f_i \\rVert = M\/\\lambda_i\n  \\]\n  \u3067\u3042\u308a, \u307e\u305f, $|\\psi_i(x)| \\leq \\lVert \\psi_i \\rVert \\lVert x \\rVert \\leq M$\u304b\u3064\n  \\[\n  |\\lambda_i f_i(x) - \\psi_i(x)| \\leq \\lVert f_i \/ \\lVert f_i \\rVert - \\psi_i \\rVert \\lVert x \\rVert < M \\delta\n  \\]\n  \u3088\u308a$| h_i(\\lambda_i f_i(x)) - h_i(\\psi_i(x)) | < \\varepsilon\/3n$\u3068\u306a\u308b. \u3088\u3063\u3066, \u4e09\u89d2\u4e0d\u7b49\u5f0f\u3088\u308a, \n  \\[\n  \\begin{aligned}\n  &\\left|h(x) - \\sum_{i=0}^n \\sum_{j=0}^{N_i} a_{i,j} g_{i,j}(f_i(x)) \\right| \\\\\n  &\\leq \\left|h(x) - \\sum_{i=0}^n h_i(\\psi_i(x))\\right| + \\left|\\sum_{i=0}^n h_i(\\psi_i(x)) - \\sum_{i=0}^n h_i(\\lambda_i f_i(x)\\right| \\\\\n  &~~~~ + \\left| \\sum_{i=0}^n h_i(\\lambda_i f_i(x)) - \\sum_{i=0}^n \\sum_{j=0}^{N_i} a_{i,j} g_{i,j}(f_i(x)) \\right| \\\\\n  &\\leq \\varepsilon\/3 + \\varepsilon\/3 + \\varepsilon\/3 = \\varepsilon.\n  \\end{aligned}\n  \\]\n\\end{proof}\n\\begin{cor}\n  \u4e0a\u306e\u5b9a\u7406\u306f$\\mathcal{F} \\subset X^*$\u304c\u5358\u4f4d\u7403\u9762\u306b\u304a\u3044\u3066\u4f5c\u7528\u7d20\u30ce\u30eb\u30e0\u306b\u95a2\u3057\u3066\u7a20\u5bc6\u3067\u3042\u308b\u3068\u3044\u3046\u4eee\u5b9a\u306e\u3082\u3068\u3067\u6210\u308a\u7acb\u3064. \n\\end{cor}\n\\begin{cor}\n  $X \\neq \\{0\\}$\u3092\u30ce\u30eb\u30e0\u7a7a\u9593\u3068\u3057, $\\mathcal{F} \\subset X^*$\u3092\u4e0a\u306e\u5b9a\u7406\u3068\u540c\u69d8\u306e\u3082\u306e\u3068\u3059\u308b. \u307e\u305f, $g \\in C(\\mathbb{R})$\u3068\u3057, $A \\subset \\mathbb{R}$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \n  \\[\n  \\mathcal{G} = \\{ g_{a}: \\mathbb{R} \\ni t \\mapsto g(t + a) \\in \\mathbb{R} \\mid a \\in A\\}\n  \\]\n  \u304c$C(\\mathbb{R})$\u306b\u304a\u3044\u3066\u57fa\u790e\u7684\u3067\u3042\u308b\u306a\u3089\u3070, $\\mathcal{G} \\circ \\mathcal{F} = \\{ g_a \\circ f \\mid a \\in A, f \\in \\mathcal{F}\\}$\u306f$C(X)$\u306b\u304a\u3044\u3066\u57fa\u790e\u7684\u3067\u3042\u308b. \n\\end{cor}\n\u3053\u306e\u7cfb\u306b\u95a2\u9023\u3057\u3066mean-periodic\u95a2\u6570\u306b\u3064\u3044\u3066\u7d39\u4ecb\u3059\u308b. \n\\begin{defn}\n   \u95a2\u6570$f \\in C(\\mathbb{R}^r)$\u304c mean-periodic \u3067\u3042\u308b\u3068\u306f, \n   \\[\n   \\mathcal{N}_r(f,\\{1\\},\\mathbb{R}) = \\mathrm{span}\\{x \\mapsto f(x+a) \\mid a \\in \\mathbb{R}^r \\}\n   \\]\n   \u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u306a\u3044\u3053\u3068\u3092\u3044\u3046. \n\\end{defn}\n\\begin{thm}[Schwartz 1947 \\cite{Schwartz1947} p.907] \\\\\n  $\\Psi \\in C(\\mathbb{R})$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u3042\u308b$1\\leq p < \\infty$\u306b\u3064\u3044\u3066$\\Psi \\in L^p(\\mathbb{R})$\u3067\u3042\u308b\u304b, \u307e\u305f\u306f$\\Psi$\u306f\u6709\u754c\u3067\u975e\u5b9a\u6570\u304b\u3064$x \\rightarrow \\infty$\u304b$x \\rightarrow -\\infty$\u3068\u3057\u305f\u3068\u304d\u306b\u6975\u9650\u5024\u3092\u6301\u3064\u306a\u3089\u3070, $\\Psi$\u306f mean-periodic\u3067\u306a\u3044. \n\\end{thm}\n\u3053\u308c\u3088\u308a\u4e0a\u306e\u5b9a\u7406\u306e\u4eee\u5b9a\u3092\u307f\u305f\u3059$\\Psi$\u306b\u3064\u3044\u3066\u306f, \u4efb\u610f\u306e\u5b9f\u6570$\\lambda \\neq 0$\u306b\u3064\u3044\u3066, $\\mathbb{N}(\\Psi,\\{\\lambda\\},\\mathbb{R})$\u306f$C(\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \u3055\u3089\u306b, Lin\u3068Pinkus\u306e\u7d50\u679c(\u5b9a\u7406\\ref{LinAndPinkus})\u3092\u7528\u3044\u308b\u3068\u6b21\u306e\u547d\u984c\u306e\u6210\u7acb\u3082\u308f\u304b\u308b. \n\\begin{prop}\n  $\\Psi$\u306f\u4e0a\u306e\u5b9a\u7406\u306e\u4eee\u5b9a\u3092\u307f\u305f\u3059\u3082\u306e\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $A \\subset \\mathbb{R}^r$\u4e0a\u3067$0$\u3092\u53d6\u308b\u975e\u81ea\u660e\u306a$r$\u5909\u6570\u6589\u6b21\u591a\u9805\u5f0f\u304c\u5b58\u5728\u3057\u306a\u3044\u306a\u3089\u3070, \n  \\[\n  \\mathcal{N}_{r}(\\Psi,A,\\mathbb{R}) = \\mathrm{span}\\{\\mathbb{R}^r \\ni x \\mapsto \\Psi(\\ip<{x,v}> + b) \\mid v \\in A, b \\in \\mathbb{R}\\}\n  \\]\n  \u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{prop}\n\u3053\u306e\u547d\u984c\u3092\u4e8c\u6bb5\u968e\u306b\u5206\u3051\u3066\u8a3c\u660e\u3057\u3088\u3046. \n\\begin{lem}\n  $X$\u3092\u30ce\u30eb\u30e0\u7a7a\u9593\u3068\u3057, $E \\subset C(\\mathbb{R})$\u306f$C(\\mathbb{R})$\u306b\u304a\u3044\u3066\u57fa\u790e\u7684\u3067\u3042\u308b\u3068\u3059\u308b. \u307e\u305f, $\\mathcal{F} \\subset X^*$\u306f$C(\\mathbb{R}) \\circ \\mathcal{F}$\u304c$C(X)$\u306b\u304a\u3044\u3066\u57fa\u790e\u7684\u306a\u3082\u306e\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $\\{g \\circ f \\mid g \\in E, f \\in \\mathcal{F}\\}$\u306f$C(X)$\u306b\u304a\u3044\u3066\u57fa\u790e\u7684\u3067\u3042\u308b. \n\\end{lem}\n\\begin{proof}\n  $K \\subset X$\u3092\u7a7a\u3067\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u3068\u3057, \u4efb\u610f\u306b$\\varphi \\in C(X)$\u3068$\\varepsilon > 0$\u3092\u53d6\u308b. \n  \u4eee\u5b9a\u3088\u308a, $\\beta_i \\in \\mathbb{R}, g_i \\in C(\\mathbb{R}), f_i \\in \\mathcal{F} ~(i=1,\\ldots,q)$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\sup_{x \\in K} |\\sum_{i=1}^q \\beta_i g_i(f_i(x)) - \\varphi(x)| < \\varepsilon\/2\n  \\]\n  \u3068\u306a\u308b. \u5404$i$\u306b\u3064\u3044\u3066$f_i$\u306f\u9023\u7d9a\u3067\u3042\u308b\u304b\u3089$f_i(K) \\subset \\mathbb{R}$\u306f\u30b3\u30f3\u30d1\u30af\u30c8\u3067\u3042\u308b. \u305d\u3053\u3067, \u4eee\u5b9a\u3088\u308a\u5404$i$\u306b\u3064\u3044\u3066$\\alpha_{i,j} \\in \\mathbb{R}, g_{i,j} \\in E ~(j=1,\\ldots,m_i)$\u304c\u5b58\u5728\u3057\u3066\n  \\[\n  \\sup_{y \\in f_i(K)} |\\sum_{j=1}^{m_i} \\alpha_{i,j} g_{i,j}(y) - \\beta_i g_i(y)| < \\varepsilon\/(2q)\n  \\]\n  \u3068\u306a\u308b. \u3059\u308b\u3068, \u4efb\u610f\u306e$x \\in K$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  &\\left| \\sum_{i=1}^q \\beta_i g_i(f_i(x)) - \\sum_{i=1}^q \\sum_{j=1}^{m_i} \\alpha_{i,j}  g_{i,j}(f_i(x)) \\right| \\\\\n  &\\leq \\sum_{i=1}^q \\left| \\beta_i g_i(f_i(x)) - \\sum_{j=1}^{m_i} \\alpha_{i,j}  g_{i,j}(f_i(x)) \\right| \\\\\n  &\\leq \\sum_{i=1}^q \\sup_{y \\in f_i(K)} \\left|\\beta_i g_i(y) - \\sum_{j=1}^{m_i} \\alpha_{i,j} g_{i,j}(y)\\right| \\\\\n  &< \\varepsilon\/2\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b\u304b\u3089, $\\sum_{i=1}^q \\sum_{j=1}^{m_i} \\alpha_{i,j}  g_{i,j} \\circ f_i$\u304c\u6c42\u3081\u308b\u3082\u306e\u3067\u3042\u308b. \n\\end{proof}\n\\begin{prop}\n  $\\Psi \\in C(\\mathbb{R})$\u3068\u3057, $B \\subset \\mathbb{R}$\u3068\u3059\u308b. \u307e\u305f, \u96c6\u5408\n  \\[\n  S := \\{ \\mathbb{R} \\ni t \\mapsto \\Psi(t+b) \\in \\mathbb{R} \\mid b \\in B \\}\n  \\]\n  \u306f$C(\\mathbb{R})$\u306b\u304a\u3044\u3066\u57fa\u790e\u7684\u3067\u3042\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $A \\subset \\mathbb{R}^r$\u4e0a\u3067$0$\u3092\u53d6\u308b\u975e\u81ea\u660e\u306a$r$\u5909\u6570\u6589\u6b21\u591a\u9805\u5f0f\u304c\u5b58\u5728\u3057\u306a\u3044\u306a\u3089\u3070, \n  \\[\n  \\mathcal{N}_{r}(\\Psi,A,B) = \\mathrm{span}\\{\\mathbb{R}^r \\ni x \\mapsto \\Psi(\\ip<{x,v}> + b) \\mid v \\in A, b \\in B\\}\n  \\]\n  \u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{prop}\n\\begin{proof}\n  \u4eee\u5b9a\u3068Lin\u3068Pinkus \u306e\u7d50\u679c\u3088\u308a$C(\\mathbb{R}) \\circ \\{x \\mapsto \\ip<{x,v}> \\mid v \\in A\\}$\u304c\u57fa\u790e\u7684\u3067\u3042\u308b\u306e\u3067\u3059\u3050\u4e0a\u306e\u88dc\u984c\u304b\u3089\u660e\u3089\u304b\u3067\u3042\u308b. \n\\end{proof}\n\n\u3055\u3066, \u672c\u7bc0\u306e\u5192\u982d\u3067\u300cChui\u3068Li \u306e\u7d50\u679c\u3092\u81ea\u7136\u306b\u30ce\u30eb\u30e0\u7a7a\u9593$X$\u4e0a\u306e\u9023\u7d9a\u95a2\u6570\u7a7a\u9593$C(X)$\u306b\u62e1\u5f35\u3057\u305f\u300d\u3068\u8ff0\u3079\u305f. \u304d\u3061\u3093\u3068\u62e1\u5f35\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u3092\u4ee5\u4e0b\u3067\u78ba\u8a8d\u3057\u3066\u304a\u304f. \n\u3064\u307e\u308a, $X=\\mathbb{R}^s$\u3068\u304a\u304f\u3068\u304d, $\\mathcal{F} = \\{x \\mapsto \\ip<{x,v}> \\mid v \\in \\mathbb{Z}^s \\}$\u3068\u304a\u304f\u3068, \n  \\[\n  \\{ f\/\\lVert f \\rVert \\mid f \\in \\mathcal{F}, f \\neq 0 \\}\n  \\]\n  \u304c$X^*$\u306e\u5358\u4f4d\u7403\u9762$\\{f \\in X^* \\mid \\lVert f \\rVert = 1\\}$\u306b\u304a\u3044\u3066\u4f5c\u7528\u7d20\u30ce\u30eb\u30e0$\\lVert \\cdot \\rVert$\u306b\u95a2\u3057\u3066\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u304b\u3081\u3066\u304a\u304f. \n\\begin{lem}\n  $X,Y$\u3092\u4f4d\u76f8\u7a7a\u9593\u3068\u3057, $f:X \\rightarrow Y$\u3092\u5168\u5c04\u9023\u7d9a\u5199\u50cf\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $A \\subset X$\u304c\u7a20\u5bc6\u3067\u3042\u308b\u306a\u3089\u3070, $f(A) \\subset Y$\u306f\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{lem}\n\\begin{proof}\n  $y \\in Y$\u3092\u3068\u308a, $y$\u306e\u8fd1\u508d$V$\u3092\u4efb\u610f\u306b\u53d6\u308b. $f$\u306f\u5168\u5c04\u306a\u306e\u3067$x \\in X$\u3067$y = f(x)$\u3092\u307f\u305f\u3059\u3082\u306e\u304c\u53d6\u308c\u308b. \u3059\u308b\u3068, $f^{-1}(V)$\u306f$x$\u306e\u8fd1\u508d\u3068\u306a\u308b. \u305d\u3053\u3067, \u4eee\u5b9a\u3088\u308a$a \\in A$\u3067$a \\in f^{-1}(V)$\u306a\u308b\u3082\u306e\u304c\u53d6\u308c\u308b. \u3057\u305f\u304c\u3063\u3066$f(a) \\in V$\u3067\u3042\u308b\u304b\u3089, $f(A) \\cap V \\neq \\emptyset$\u3067\u3042\u308b. \u3053\u308c\u306f$f(A)$\u304c$Y$\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b. \n\\end{proof}\n\\begin{prop}\n  $X=\\mathbb{R}^s$\u3068\u304a\u304f\u3068\u304d, $\\mathcal{F} = \\{x \\mapsto \\ip<{x,v}> \\mid v \\in \\mathbb{Z}^s \\}$\u3068\u304a\u304f\u3068, \n  \\[\n  \\{ f\/\\lVert f \\rVert \\mid f \\in \\mathcal{F}, f \\neq 0 \\}\n  \\]\n  \u306f$X^*$\u306e\u5358\u4f4d\u7403\u9762$\\{f \\in X^* \\mid \\lVert f \\rVert = 1\\}$\u306b\u304a\u3044\u3066\u4f5c\u7528\u7d20\u30ce\u30eb\u30e0$\\lVert \\cdot \\rVert$\u306b\u95a2\u3057\u3066\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{prop}\n\\begin{proof}\n  \u30b7\u30e5\u30ef\u30eb\u30c4\u306e\u4e0d\u7b49\u5f0f\u304b\u3089$\\mathbb{R}^s \\ni x \\mapsto \\ip<{x,u}> \\in \\mathbb{R}$\u306e\u4f5c\u7528\u7d20\u30ce\u30eb\u30e0\u306f$\\lVert u \\rVert$\u3068\u4e00\u81f4\u3059\u308b\u306e\u3067, \n  \\[\n  \\{f\/ \\lVert f \\rVert \\mid f \\in \\mathcal{F} , f \\neq 0 \\}\n  \\]\n  \u304c$\\{\\mathbb{R}^s \\ni x \\mapsto \\ip<{x,u}> \\in \\mathbb{R} \\mid u \\in \\mathbb{R}^s, \\lVert u \\rVert = 1 \\}$\u306b\u304a\u3044\u3066\u4f5c\u7528\u7d20\u30ce\u30eb\u30e0\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044. \u305d\u3057\u3066, \u3053\u306e\u3053\u3068\u3092\u793a\u3059\u306b\u306f, \n  \\[\n  S = \\{m\/\\lVert m \\rVert \\mid m \\in \\mathbb{Z}, z \\neq 0\\}\n  \\]\n  \u304c$U = \\{ u \\in \\mathbb{R}^s \\mid \\lVert u \\rVert = 1 \\}$\u306b\u304a\u3044\u3066\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u8ddd\u96e2\u306b\u95a2\u3057\u3066\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u5341\u5206\u3067\u3042\u308b. \n  \u3044\u307e, \u5199\u50cf$\\varphi$\u3092\n  \\[\n  \\varphi:\\mathbb{R}^s \\setminus \\{0\\} \\rightarrow U, ~ \\varphi(x) = x\/\\lVert x \\rVert\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b\u3068, \u3053\u308c\u306f\u9023\u7d9a\u304b\u3064\u5168\u5c04\u3067\u3042\u308b. \u305d\u3057\u3066, $\\mathbb{Q}^s \\setminus \\{0\\}$\u306f$\\mathbb{R}^s \\setminus \\{0\\}$\u306b\u304a\u3044\u3066\u7a20\u5bc6\u3067\u3042\u308b\u304b\u3089, \u3059\u3050\u4e0a\u306e\u88dc\u984c\u3088\u308a\n  \\[\n  \\varphi(\\mathbb{Q}^s \\setminus \\{0\\}) = \\{ q\/\\lVert q \\rVert \\mid q \\in \\mathbb{Q}^s , q \\neq 0 \\}\n  \\]\n  \u306f$U$\u306b\u304a\u3044\u3066\u7a20\u5bc6\u3067\u3042\u308b. \u3057\u304b\u3082, $\\varphi(\\mathbb{Q}^s \\setminus \\{0\\}) = S$\u3067\u3042\u308b. \u5b9f\u969b, $q \\in \\mathbb{Q}^s \\setminus \\{0\\}$\u3092\u53d6\u308b\u3068, $n \\in \\mathbb{Z}^s$\u3068$m \\in \\mathbb{Z}$\u3092\u4f7f\u3063\u3066$q = n\/m $\u3068\u8868\u305b\u308b\u306e\u3067, $q\/\\lVert q \\rVert = n\/\\lVert n \\rVert \\in S$\u3068\u306a\u308b. \u9006\u5411\u304d\u306e\u5305\u542b\u95a2\u4fc2\u306f\u660e\u3089\u304b\u3067\u3042\u308b. \u3088\u3063\u3066, $S$\u306f$U$\u306b\u304a\u3044\u3066\u7a20\u5bc6\u3067\u3042\u308b. \u4ee5\u4e0a\u3067\u7cfb\u306f\u793a\u3055\u308c\u305f. \n\\end{proof}\n\u306a\u304a, \u3053\u306e\u547d\u984c\u304b\u3089\u6b21\u306e\u3053\u3068\u3082\u76f4\u3061\u306b\u308f\u304b\u308b. \n\\begin{prop}\n  $\\mathcal{G} \\subset C(\\mathbb{R})$\u304c\u57fa\u790e\u7684\u3067\u3042\u308b\u3068\u3059\u308b\u3068, \n  \\[\n  \\{ \\mathbb{R}^s \\ni x \\mapsto g(\\ip<{x,v}>) \\in \\mathbb{R} \\mid g \\in \\mathcal{G}, v \\in \\mathbb{Z}^s \\}\n  \\]\n  \u306f$C(\\mathbb{R}^s)$\u306b\u304a\u3044\u3066\u57fa\u790e\u7684\u3067\u3042\u308b. \n\\end{prop}\n\n\n\n\\subsection{Hornik\u306e\u7d50\u679c}\n\u672c\u7bc0\u3067\u306fHornik\u306e\u7d50\u679c, \u3064\u307e\u308a, \u6d3b\u6027\u5316\u95a2\u6570$\\Psi$\u304csquashing\u95a2\u6570\u3067\u3042\u308b\u3068\u304d\u306b$\\Sigma^r(\\Psi)$\u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u306e\u8a3c\u660e\u3092\u4e0e\u3048\u308b. \u305f\u3060\u3057, squashing\u95a2\u6570\u306e\u5b9a\u7fa9\u306f\u6b21\u306e\u901a\u308a\u3067\u3042\u308b. \n\\begin{defn}[squashing\u95a2\u6570] \\\\\n \u95a2\u6570 $\\Psi : \\mathbb{R} \\rightarrow [0,1]$\u304csquashing\u95a2\u6570\u3067\u3042\u308b\u3068\u306f, $\\Psi$\u304c\u5358\u8abf\u975e\u6e1b\u5c11\u3067\u3042\u308a, \n \\[\n \\Psi(t) \\rightarrow\n \\begin{cases}\n 1 & (t \\rightarrow +\\infty)\\\\\n 0 & (t \\rightarrow -\\infty)\n \\end{cases}\n \\]\n \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u3044\u3046. \u3064\u307e\u308a, \u5358\u8abf\u975e\u6e1b\u5c11\u306asigmoidal\u95a2\u6570\u3092squashing\u95a2\u6570\u3068\u3044\u3046. \n\\end{defn}\n\\begin{rem}\n  squashing\u95a2\u6570\u306fBorel\u53ef\u6e2c\u3067\u3042\u308b. \u306a\u305c\u306a\u3089,  \u5358\u8abf\u975e\u6e1b\u5c11\u95a2\u6570\u306fBorel\u53ef\u6e2c\u3060\u304b\u3089\u3067\u3042\u308b. \u5b9f\u969b, $f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3092\u5358\u8abf\u975e\u6e1b\u5c11\u95a2\u6570\u3068\u3059\u308b\u3068\u304d, \u4efb\u610f\u306e$a \\in \\mathbb{R}$\u306b\u5bfe\u3057\u3066, $x = \\sup\\{y \\in \\mathbb{R} \\mid f(y) \\leq a\\}$\u3068\u304a\u304f\u3068, \n  \\[\n  f^{-1}((-\\infty,a]) = (-\\infty,x) ~\\mbox{or}~ (-\\infty,x] \\in \\mathcal{B}_{\\mathbb{R}}\n  \\]\n  \u3068\u306a\u308b\u304b\u3089, $f$\u306fBorel\u53ef\u6e2c\u3067\u3042\u308b. \n\\end{rem}\nsquashing\u95a2\u6570\u306b\u306f\u9023\u7d9a\u6027\u304c\u8981\u6c42\u3055\u308c\u3066\u3044\u306a\u3044\u3053\u3068\u306b\u6ce8\u610f\u3055\u308c\u305f\u3044. Hornik\u306f\u9023\u7d9a\u6027\u306e\u4eee\u5b9a\u3092\u306a\u304f\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u304b\u3068\u3044\u3046\u3053\u3068\u306b\u95a2\u5fc3\u304c\u3042\u3063\u305f\u3088\u3046\u3067\u3042\u308b\\cite{Hornik}.\nLeshno et al.\u306e\u7d50\u679c\u306e\u7bc0\u3067\u7d39\u4ecb\u3057\u305fHornik\u306e\u7d50\u679c\u306b\u3082\u3053\u306e\u59ff\u52e2\u306f\u8868\u308c\u3066\u3044\u308b. \n\n\u3055\u3066, \u307e\u305a\u6b21\u306e$\\Sigma \\Pi$-\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306b\u3064\u3044\u3066\u8003\u5bdf\u3057\u3066\u3044\u304f. \n\\begin{thm}\\label{MainTheorem1}\n  $G : \\mathbb{R} \\rightarrow \\mathbb{R}$\u3092\u5b9a\u6570\u3067\u306a\u3044\u9023\u7d9a\u95a2\u6570\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \n  \\[\n  \\Sigma \\Pi^r(G) \n  := \\left\\{ f:\\mathbb{R}^r \\rightarrow \\mathbb{R} ~\\vline~ \n  \\begin{aligned}\n  &f = \\sum_{j}^{q} \\beta_j \\prod_{k=1}^{l_j}  G \\circ A_{j,k} \\\\\n  &\\beta_j \\in \\mathbb{R}, A_{j,k} \\in \\mathbb{A}^r , l_j \\in \\mathbb{N},  q \\in \\mathbb{N}\n  \\end{aligned}\n  \\right\\}\n  \\]\n  \u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{thm}\n\\begin{proof}\n  \u4efb\u610f\u306b\u7a7a\u3067\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}^r$\u3092\u53d6\u308b. \u3059\u308b\u3068$\\Sigma \\Pi^r(G)_K$\u306f$C(K)$\u306e\u90e8\u5206\u4ee3\u6570\u3067\u3042\u308b. \n  \u5b9f\u969b, \u7dda\u5f62\u90e8\u5206\u7a7a\u9593\u3067\u3042\u308b\u3053\u3068\u306f\u660e\u3089\u304b\u3067\u3042\u308b. \n  \u307e\u305f, $f,g \\in \\Sigma \\Pi^r(G)_K$\u306a\u3089\u3070$f g \\in \\Sigma \\Pi^r(G)_K$\u3067\u3042\u308b\u3053\u3068\u3082\u6709\u9650\u500b\u306e\u7d44$(i,j)$\u3092\u4e00\u5217\u5316\u3059\u308b\u3053\u3068\u3067\u793a\u3055\u308c\u308b. \n  \u3044\u307e, $G$\u306f\u5b9a\u6570\u3067\u306a\u3044\u306e\u3067$G(a) \\neq 0$\u306a\u308b$a \\in \\mathbb{R}$\u304c\u53d6\u308c\u308b. \n  \u3053\u306e\u3068\u304d, $A(x) := 0^{\\mathrm{T}} x + a = a$\u3068\u5b9a\u7fa9\u3059\u308b\u3068, $G \\circ A \/ G(a) \\in \\Sigma \\Pi^r(G)_K$\u3068\u306a\u308a, $G(A(x))\/G(a) = 1 ~~(\\forall x \\in K)$\u3068\u306a\u308b. \n  \u6b21\u306b, $x,y \\in K , ~ x \\neq y$\u3092\u3068\u308b. \n  $G$\u306f\u5b9a\u6570\u3067\u306a\u3044\u306e\u3067, $G(a) \\neq G(b)$\u306a\u308b$a,b \\in \\mathbb{R}$\u304c\u53d6\u308c\u308b. \n  \u3044\u307e, $A(z) = w^{\\mathrm{T}} z + \\beta$\u3067$A(x) = a, A(y) = b$\u306a\u308b\u3082\u306e\u304c\u53d6\u308c\u308b. \u5b9f\u969b, $x \\neq y$\u3086\u3048, \u3042\u308b$i$\u3067$x_i - y_i \\neq 0$\u3068\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066, $w_i = (a-b)\/(x_i-y_i), w_j = 0 ~(j \\neq i)$\u3068\u3057\u3066, $\\beta = (b x_i - a y_i)\/(x_i - y_i)$\u3068\u304a\u3051\u3070\u826f\u3044. \n  \u3053\u306e\u3068\u304d, $G(A(x)) = a \\neq b = G(A(y))$\u3067\u3042\u308b. \n  \u3088\u3063\u3066, Stone-Weierstrass\u306e\u5b9a\u7406\u306b\u3088\u3063\u3066, $\\Sigma \\Pi^r(G)_K$\u306f$C(K)$\u306b\u304a\u3044\u3066\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n  \u4ee5\u4e0a\u304b\u3089, $\\Sigma \\Pi^r(G)$\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{proof}\n\\begin{lem}\\label{squashingFuncUniAppro}\n  $\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3092squashing\u95a2\u6570\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e\u9023\u7d9a\u306asquashing\u95a2\u6570$F$\u3068$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, \u3042\u308b$H \\in \\Sigma^1(\\Psi)$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\sup_{x \\in \\mathbb{R}} |F(x) - H(x)| < \\varepsilon.\n  \\]\n\\end{lem}\n\\begin{proof}\n  \u4efb\u610f\u306b$\\varepsilon > $\u3092\u53d6\u308b. $\\varepsilon < 1$\u3068\u3057\u3066\u3088\u3044. $\\beta_j \\in \\mathbb{R}$\u3068$A_j \\in \\mathbb{A}^1$\u3067, \n  \\[\n  \\sup_{x \\in \\mathbb{R}} |F(x) - \\sum_{j=0}^{Q-1} \\beta_j \\Psi(A_j(x))| < \\varepsilon\n  \\]\n  \u306a\u308b\u3082\u306e\u3092\u898b\u3064\u3051\u305f\u3044. \u305d\u3053\u3067$Q \\in \\mathbb{N}$\u3092$1\/Q < \\varepsilon\/2$\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u53d6\u308b. \u305d\u3057\u3066, $\\beta_j = 1\/Q$\u3068\u304a\u304f. \u3044\u307e$\\Psi$\u306fsquashing\u95a2\u6570\u306a\u306e\u3067$M > 0$\u3067\n  \\[\n  \\Psi(-M) < \\frac{1}{Q},~~\\Psi(M) > 1 - \\frac{1}{Q}\n  \\]\n  \u306a\u308b\u3082\u306e\u304c\u53d6\u308c\u308b. \u307e\u305f, $F$\u306f\u9023\u7d9a\u306asquashing\u95a2\u6570\u306a\u306e\u3067\n  \\[\n  \\begin{aligned}\n  &r_j = \\sup\\{x \\in \\mathbb{R} \\mid F(x) = j\/Q \\} ~~~(j=1,\\ldots,Q-1) \\\\\n  &r_Q = \\sup \\{x \\in \\mathbb{R} \\mid F(x) = 1 - 1\/(2Q) \\}\n  \\end{aligned}\n  \\]\n \u304c\u5b58\u5728\u3059\u308b. \u305d\u3057\u3066, $r_0 \\in (-\\infty,r_1)$\u3092\u4efb\u610f\u306b\u53d6\u308b. \u3055\u3066, $r < s$\u306b\u5bfe\u3057\u3066$A_{r,s} \\in \\mathbb{A}^1$\u3092$A_{r,s}(r) = -M, A_{r,s}(s) = M$\u306a\u308b\u3082\u306e\u3068\u3059\u308b(\u3053\u308c\u306f\u4e00\u610f\u306b\u5b9a\u307e\u308b). \u3053\u306e\u3068\u304d, $A_j = A_{r_j,r_{j+1}}$\u3068\u3059\u308b\u3068, \n \\[\n H = \\sum_{j=0}^{Q-1} \\beta_j \\Psi \\circ A_j ~\\in \\Sigma^1(\\Psi)\n \\]\n \u304c\u6c42\u3081\u308b\u3082\u306e\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u5b9f\u969b, $x \\in (-\\infty,r_0]$\u306e\u5834\u5408\u306f, $0 \\leq F(x) \\leq 1\/Q$\u3067\u3042\u308a, \u304b\u3064$\\Psi$\u304csquashing\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u3088\u308a, \n \\[\n 0 \\leq H(x) \\leq \\sum_{j=0}^{Q-1} \\frac{1}{Q} \\Psi(-M) \\leq \\frac{1}{Q}\n \\]\n \u3068\u306a\u308b. \u3088\u3063\u3066, $|F(x) - H(x)| \\leq 1\/Q < \\varepsilon$\u3068\u306a\u308b. \u6b21\u306b, $x \\in (r_0,r_1]$\u306e\u5834\u5408\u3092\u8003\u3048\u308b. \u3053\u306e\u3068\u304d, $0 \\leq F(x) \\leq 1\/Q$\u3067\u3042\u308a, \n \\[\n \\begin{aligned}\n 0  \\leq H(x) \n &\\leq \\frac{1}{Q} + \\sum_{j=1}^{Q-1} \\frac{1}{Q} \\Psi(-M) \\\\\n &\\leq \\frac{1}{Q} + \\frac{Q-1}{Q}\\frac{1}{Q} \\\\\n &\\leq \\frac{1}{Q} + \\frac{1}{Q}\n \\end{aligned}\n \\]\n \u3068\u306a\u308b\u306e\u3067, $|F(x)-H(x)| \\leq 1\/Q + 1\/Q \\leq \\varepsilon$\u3068\u306a\u308b. \u6b21\u306b$x \\in (r_j,r_{j+1}] ~~(j=1,\\ldots,Q-2)$\u306e\u5834\u5408\u3092\u8003\u3048\u308b. \u3053\u306e\u3068\u304d, $r_j$\u306e\u5b9a\u7fa9\u3088\u308a\n \\[\n \\frac{j}{Q} \\leq F(x) \\leq \\frac{j+1}{Q}\n \\]\n \u3067\u3042\u308b. \u307e\u305f, \n \\[\n \\begin{aligned}\n H(x) \\geq \\sum_{k=0}^{j-1} \\frac{1}{Q} \\Psi(M) \\geq \\frac{j}{Q} - \\frac{j}{Q^2} \\geq \\frac{j-1}{Q}\n \\end{aligned}\n \\]\n \u304b\u3064\n \\[\n \\begin{aligned}\n H(x) \n &\\leq \\sum_{k=0}^{j} \\frac{1}{Q} + \\sum_{k=j+1}^{Q-1} \\frac{1}{Q} \\Psi(-M) \\\\\n &\\leq \\frac{j+1}{Q} + \\frac{1}{Q} = \\frac{(j+2}{Q}\n \\end{aligned}\n \\]\n \u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, $|F(x)-H(x)| \\leq 2\/Q \\leq \\varepsilon$\u3068\u306a\u308b. \u6b21\u306b$x \\in (r_{Q-1},r_Q]$\u306e\u5834\u5408\u3092\u8003\u3048\u308b. \u3053\u306e\u3068\u304d, \n \\[\n 1 - \\frac{1}{Q} = \\frac{Q-1}{Q} \\leq F(x) \\leq 1 - \\frac{1}{2Q} \n \\]\n \u3067\u3042\u308b. \u307e\u305f, \n \\[\n H(x) \\geq \\sum_{j=0}^{Q-2} \\frac{1}{Q} \\Psi(M) \\geq  \\frac{Q-1}{Q} - \\frac{Q-1}{Q^2} \\geq \\frac{Q-2}{Q} = 1 - \\frac{2}{Q}\n \\]\n \u304b\u3064, \n \\[\n H(x) \\leq \\sum_{j=0}^{Q-1} \\frac{1}{Q} = 1\n \\]\n \u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, $|F(x)-H(x)| \\leq \\max\\{1\/Q,2\/Q-1\/(2Q)\\} \\leq \\varepsilon$\u3068\u306a\u308b. \u6700\u5f8c\u306b, $x \\in (r_Q,\\infty)$\u306e\u5834\u5408\u3092\u8003\u3048\u308b. \u3053\u306e\u3068\u304d, \n $1 - 1\/(2Q) \\leq F(x) \\leq 1$\u3067\u3042\u308a, \n \\[\n 1 \\geq H(x) \\geq \\sum_{j=0}^{Q-1} \\frac{1}{Q} \\Psi(M) \\geq 1 - \\frac{1}{Q}\n \\]\n \u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, $|F(x)-H(x)| \\leq 1\/Q \\leq \\varepsilon$\u3068\u306a\u308b. \n\\end{proof}\n\\begin{thm}\\label{MainTheorem3}\n  $\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3092squashing\u95a2\u6570\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $\\Sigma \\Pi^r(\\Psi)$\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{thm}\n\\begin{proof}\n  \u9023\u7d9a\u306asquashing\u95a2\u6570$F$\u306b\u5bfe\u3057\u3066, $\\Sigma \\Pi^r(\\Psi)$\u304c$\\Sigma \\Pi^r(F)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u5341\u5206\u3067\u3042\u308b. \n  \u306a\u305c\u306a\u3089, \u305d\u308c\u304c\u793a\u3055\u308c\u308c\u3070\u5b9a\u7406\\ref{MainTheorem1}\u306b\u3088\u308a$\\Sigma \\Pi^r(F)$\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u306e\u3067, $\\Sigma \\Pi^r(\\Psi)$\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3068\u306a\u308b. \n  \n  \u3055\u3066, \u3053\u306e\u3053\u3068\u3092\u793a\u3059\u305f\u3081\u306b\u306f, $\\prod_{k=1}^{l} F \\circ A_k$\u306e\u5f62\u306e\u95a2\u6570\u3092$\\Sigma \\Pi^r(\\Psi)$\u306e\u5143\u3067\u4e00\u69d8\u8fd1\u4f3c\u3067\u304d\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044. \n  \u305d\u3053\u3067, $\\varepsilon > 0$\u3092\u4efb\u610f\u306b\u53d6\u308b. \u7a4d$\\prod_{k=1}^{l} : [0,1]^l \\ni (a_1,\\ldots,a_l) \\mapsto \\prod_{k=1}^{l} a_l \\in \\mathbb{R}$\u306f\u9023\u7d9a\u3067$[0,1]^l$\u306f\u30b3\u30f3\u30d1\u30af\u30c8\u306a\u306e\u3067, \u3042\u308b$\\delta$\u304c\u5b58\u5728\u3057\u3066\u4efb\u610f\u306e$a_k,b_k \\in [0,1]$\u306b\u5bfe\u3057\u3066, $|a_k - b_k| < \\delta ~~(k=1,\\ldots,l)$\u306a\u3089\u3070$|\\prod_{k=1}^l a_k - \\prod_{k=1}^l b_k| < \\varepsilon$\u3068\u306a\u308b. \n  \u4e00\u65b9, \u88dc\u984c\\ref{squashingFuncUniAppro}\u306b\u3088\u308a, \u3042\u308b$\\beta_j \\in \\mathbb{R}$\u3068$A_j^1 \\in \\mathbb{A}^1$\u304c\u5b58\u5728\u3057\u3066, $H = \\sum_{j=1}^{m} \\beta_j \\Psi \\circ A_j^1$\u306b\u3064\u3044\u3066\n  \\[\n  \\sup_{x \\in \\mathbb{R}} |F(x) - H(x)| < \\delta\n  \\]\n  \u304c\u6210\u308a\u7acb\u3064. \u3057\u305f\u304c\u3063\u3066, \n  \\[\n  \\sup_{x \\in \\mathbb{R}^r} \\left| \\prod_{k=1}^l F(A_k(x)) - \\prod_{k=1}^l H(A_k(x)) \\right| \\leq \\varepsilon\n  \\]\n  \u3068\u306a\u308b. \u3044\u307e, $A_j^1 \\circ A_k \\in \\mathbb{A}^r$\u3067\u3042\u308a, $\\Sigma \\Pi(\\Psi)$\u306f\u7a4d\u3068\u548c\u306b\u95a2\u3057\u3066\u9589\u3058\u3066\u3044\u308b\u306e\u3067, $\\prod_{k=1}^l H \\circ A_k \\in \\Sigma \\Pi(\\Psi)$\u3067\u3042\u308b. \u3053\u308c\u3067\u8a3c\u660e\u306f\u5b8c\u4e86\u3057\u305f. \n\\end{proof}\n\u6b21\u306b\u672c\u984c\u306e\u5b9a\u7406\u3092\u793a\u3057\u3066\u3044\u304f. \n\\begin{thm}[Hornik 1989 \\cite{Hornik}]\\label{MainTheorem4} \\\\\n  $\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3092squashing\u95a2\u6570\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $\\Sigma^r(\\Psi)$\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b.  \n\\end{thm}\n\u3053\u306e\u5b9a\u7406\u3092\u8a3c\u660e\u3059\u308b\u305f\u3081\u306b$2$\u3064\u88dc\u984c\u3092\u793a\u3059. \n\\begin{lem}\n  $\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3092squashing\u95a2\u6570\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$\\varepsilon,M > 0$\u306b\u5bfe\u3057\u3066, $F \\in \\Sigma^1(\\Psi)$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\sup_{x \\in [-M,M]} |F(x) - \\cos(x)| < \\varepsilon\n  \\]\n  \u304c\u6210\u7acb\u3059\u308b. \u4fbf\u5b9c\u7684\u306b\u3053\u306e\u3088\u3046\u306a$F$\u3092\u4efb\u610f\u306b\u4e00\u3064\u53d6\u308a$\\cos_{M,\\varepsilon}$\u3067\u8868\u3059\u3053\u3068\u306b\u3059\u308b. \n\\end{lem}\n\\begin{proof}\n  $n$\u3092$M \\leq (2n-1\/2)\\pi$\u3068\u306a\u308b\u3088\u3046\u306b\u53d6\u308b. \u3053\u306e\u3068\u304d, $[-M,M] \\subset [-(2n+1\/2)\\pi, (2n-1\/2)\\pi]$\u3067\u3042\u308b. \u3044\u307e, \u9023\u7d9a\u306asquashing\u95a2\u6570$g,h:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3092\n  \\[\n  g(x) \n  = \\left\\{\n  \\begin{aligned}\n  ~~~ 0 ~~~~~~~&(x \\leq -\\pi\/2) \\\\\n  \\cos(x) ~~~&(-\\pi\/2 \\leq x \\leq 0) \\\\\n  1 ~~~~~~~&(0 \\leq x)\n  \\end{aligned}\n  \\right.\n  \\]\n  \u304a\u3088\u3073, \n  \\[\n  h(x) \n  = \\left\\{\n  \\begin{aligned}\n  ~~~ 0 ~~~~~~~&(x \\leq -\\pi) \\\\\n  1+\\cos(x) ~~~&(-\\pi \\leq x \\leq -\\pi\/2) \\\\\n  1 ~~~~~~~&(-\\pi\/2 \\leq x)\n  \\end{aligned}\n  \\right.\n  \\]\n  \u3067\u5b9a\u7fa9\u3059\u308b. \u305d\u3057\u3066$g_{\\alpha}(x) = g(x - \\alpha\\pi) ~~(\\alpha \\in \\mathbb{R})$\u3068\u5b9a\u7fa9\u3059\u308b. $h_{\\alpha}$\u3082\u540c\u69d8\u306b\u5b9a\u7fa9\u3059\u308b. \u3059\u308b\u3068,  $g_{\\alpha},h_{\\alpha}$\u306f\u9023\u7d9a\u306asquashing\u95a2\u6570\u306a\u306e\u3067\u88dc\u984c\\ref{squashingFuncUniAppro}\u3088\u308a, \u4efb\u610f\u306e$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066$G_{\\alpha},H_{\\alpha} \\in \\Sigma^1(\\Psi)$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  &\\sup_{x \\in \\mathbb{R}} |g_{\\alpha}(x) - G_{\\alpha}(x)| < \\varepsilon \\\\\n  &\\sup_{x \\in \\mathbb{R}} |h_{\\alpha}(x) - H_{\\alpha}(x)| < \\varepsilon\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u4e00\u65b9, $\\tilde{h}_{\\alpha} := h_{\\alpha} - 1$\u3068\u304a\u304f\u3068, \u4efb\u610f\u306e$x \\in [-(2n+1\/2)\\pi, (2n-1\/2)\\pi]$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\cos(x) = \\sum_{j=0}^{4n-1} \\left((-1)^j g_{-2n + j}(x) + (-1)^{j+1} \\tilde{h}_{-2n + 1 + j}(x)\\right)\n  \\]\n  \u3067\u3042\u308b. \u307e\u305f, $1 \\geq \\Psi(K) > 1 - \\varepsilon$\u306a\u308b$K$\u3092\u53d6\u308a,  $\\tilde{H}_{\\alpha} :=H_{\\alpha} - \\Psi(K)$\u3068\u304a\u304f\u3068, $\\Sigma^1(\\Psi)$\u306f\u548c\u306b\u95a2\u3057\u3066\u9589\u3058\u3066\u3044\u308b\u306e\u3067$\\tilde{H}_{\\alpha} \\in \\Sigma^1(\\Psi)$\u3067\u3042\u308a, $\\sup_{x \\in \\mathbb{R}} |\\tilde{H}_{\\alpha}(x) - \\tilde{h}_{\\alpha}| < 2\\varepsilon$\u3068\u306a\u308b. \u3088\u3063\u3066, \n  \\[\n  F = \\sum_{j=0}^{4n-1} \\left((-1)^j G_{-2n + j}(x) + (-1)^{j+1} \\tilde{H}_{-2n + 1 + j}(x)\\right) ~\\in \\Sigma^1(\\Psi)\n  \\]\n  \u304c\u6c42\u3081\u308b\u3082\u306e\u3067\u3042\u308b. \u5b9f\u969b, \n \\[\n \\begin{aligned}\n \\sup_{x \\in [-M,M]} |F(x) - \\cos(x)|\n &\\leq \\sum_{j=0}^{4n-1} \\sup_{x \\in [-M,M]} |G_{-2n + j}(x) - g_{-2n + j}(x)| \\\\\n &~~~ + \\sum_{j=0}^{4n-1} \\sup_{x \\in [-M,M]} |\\tilde{H}_{-2n + 1 + j}(x)-\\tilde{h}_{-2n + 1 + j}(x)| \\\\\n &\\leq 4n\\varepsilon + 8n\\varepsilon = 12n\\varepsilon. \n \\end{aligned}\n \\]\n\\end{proof}\n\\begin{lem}\\label{CompactUniApproTheorem}\n  $g = \\sum_{j=1}^Q \\beta_j \\cos \\circ A_j , ~A_j \\in \\mathbb{A}^r$\u3068\u3059\u308b. \u307e\u305f, $\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3092squashing\u95a2\u6570\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$\\varepsilon > 0$\u3068\u4efb\u610f\u306e\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$C \\subset \\mathbb{R}^r$\u306b\u5bfe\u3057\u3066, $f \\in \\Sigma^r(\\Psi)$\u304c\u5b58\u5728\u3057\u3066, $\\sup_{x \\in C}|f(x)-g(x)| < \\varepsilon$\u3068\u306a\u308b. \n\\end{lem}\n\\begin{proof}\n  $C$\u304c\u30b3\u30f3\u30d1\u30af\u30c8\u3067\u5404$A_j$\u306f\u9023\u7d9a\u306a\u306e\u3067$A_j(C) \\subset \\mathbb{R}$\u306f\u30b3\u30f3\u30d1\u30af\u30c8\u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, $M > 0$\u3092\u9069\u5f53\u306b\u53d6\u308c\u3070$A_j(K) \\subset [-M,M] ~(j=1,\\ldots,Q)$\u3068\u306a\u308b. \u305d\u3053\u3067$K = Q\\sum_{j=1}^{Q} |\\beta_j|$\u3068\u304a\u304f\u3068, \n  \\[\n  \\sup_{x \\in C} \\left| \\sum_{j=1}^Q \\beta_j \\cos_{M,\\varepsilon\/K} (A_j(x)) - g(x) \\right| < \\varepsilon\n  \\]\n  \u3068\u306a\u308b. \u305d\u3057\u3066, $\\cos_{M,\\varepsilon\/K} \\in \\Sigma^1(\\Psi)$\u306a\u306e\u3067, $\\cos_{M,\\varepsilon\/K} \\circ A_j \\in \\Sigma^r(\\Psi)$\u3068\u306a\u308b. \u3088\u3063\u3066, $\\sum_{j=1}^Q \\beta_j \\cos_{M,\\varepsilon\/K} \\circ A_j \\in \\Sigma^r(\\Psi)$\u3067\u3042\u308b\u306e\u3067, \u3053\u308c\u3067\u88dc\u984c\u306e\u6210\u7acb\u304c\u78ba\u304b\u3081\u3089\u308c\u305f. \n\\end{proof}\n\\newtheorem*{proof1}{\u5b9a\u7406\\ref{MainTheorem4}\u306e\u8a3c\u660e.}\n\\begin{proof1}\n  $\\cos:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f\u5b9a\u6570\u3067\u306a\u3044\u9023\u7d9a\u95a2\u6570\u306a\u306e\u3067, \u5b9a\u7406\\ref{MainTheorem1}\u306b\u3088\u308a, \n  \\[\n  \\left\\{ \\sum_{j=1}^Q \\beta_j \\prod_{k=1}^{l_j} \\cos \\circ A_{j,k} \\mid Q,l_j \\in \\mathbb{N} , \\beta_j \\in \\mathbb{R}, A_{j,k} \\in \\mathbb{A}^r  \\right\\}\n  \\]\n  \u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \u3068\u3053\u308d\u304c, \u4efb\u610f\u306e$a,b \\in \\mathbb{R}$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\cos(a)\\cos(b) = \\frac{\\cos(a+b)-\\cos(a-b)}{2}\n  \\]\n  \u3067\u3042\u308a, $\\mathbb{A}^r$\u306f\u52a0\u6cd5\u3068\u30b9\u30ab\u30e9\u30fc\u500d\u306b\u95a2\u3057\u3066\u9589\u3058\u3066\u3044\u308b\u306e\u3067, \u3042\u308b$\\alpha_j \\in \\mathbb{R}, A_j \\in \\mathbb{A}^r$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\sum_{j=1}^Q \\beta_j \\prod_{k=1}^{l_j} \\cos \\circ A_{j,k} = \\sum_j \\alpha_j \\cos \\circ A_j\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, \u88dc\u984c\\ref{CompactUniApproTheorem}\u3088\u308a, $\\Sigma^r(\\Psi)$\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \\qed\n\\end{proof1}\n\n\\section{$C(\\mathbb{R}^r)$\u306b\u304a\u3051\u308b\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406\u306e\u5fdc\u7528}\n\n\\subsection{\u591a\u51fa\u529b\u30fb\u591a\u5c64\u3078\u306e\u62e1\u5f35}\n\u524d\u7ae0\u3067\u8ff0\u3079\u305f\u7d50\u679c\u306f\u3059\u3079\u3066, \u30b9\u30ab\u30e9\u30fc\n\u3092\u51fa\u529b\u3059\u308b\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306b\u95a2\u3059\u308b\u3082\u306e\u3067\u3042\u3063\u305f\u304c,  \u3053\u308c\u3089\u306f\u30d9\u30af\u30c8\u30eb\u5024\u3092\u51fa\u529b\u3059\u308b\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306b\u3064\u3044\u3066\u3082\u6210\u308a\u7acb\u3064. \u672c\u7bc0\u3067\u306f\u3053\u306e\u3053\u3068\u3092\u6b63\u78ba\u306b\u8ff0\u3079\u3088\u3046. \n\\begin{defn}\n \u95a2\u6570$G:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306b\u5bfe\u3057\u3066, \n \\[\n \\begin{aligned}\n &\\Sigma^{r,s}(G) := \\{ f:\\mathbb{R}^r \\rightarrow \\mathbb{R}^s \\mid f = \\sum_{j}^{q} \\beta_j G \\circ A_j ,~ \\beta_j \\in \\mathbb{R}^s, A_j \\in \\mathbb{A}^r , q \\in \\mathbb{N}  \\} \\\\\n &\\Sigma \\Pi^{r,s}(G) := \n \\left\\{ f:\\mathbb{R}^r \\rightarrow \\mathbb{R}^s ~\\vline~ \n \\begin{aligned}\n &f = \\sum_{j}^{q} \\beta_j \\prod_{k=1}^{l_j}  G \\circ A_{j,k} ~, \\beta_j \\in \\mathbb{R}^s \\\\\n &A_{j,k} \\in \\mathbb{A}^r , l_j \\in \\mathbb{N},  q \\in \\mathbb{N} \n \\end{aligned}\n \\right\\}\n \\end{aligned}\n \\]\n \u3068\u5b9a\u7fa9\u3059\u308b. \n\\end{defn}\n\\begin{thm}\\label{MainTheorem5}\n  \u95a2\u6570$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f$\\Sigma^r(\\Psi)$\u304c$C(\\mathbb{R}^r,\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u306b\u306a\u308b\u3082\u306e\u3068\u3059\u308b. \n  \u3053\u306e\u3068\u304d, $\\Sigma^{r,s}(\\Psi)$\u306f$C(\\mathbb{R}^r,\\mathbb{R}^s)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \u3059\u306a\u308f\u3061, \u4efb\u610f\u306b\u7a7a\u3067\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}^r$\u3092\u53d6\u308b\u3068\u304d, \u4efb\u610f\u306e$g \\in C(\\mathbb{R}^r,\\mathbb{R}^s)$\u3068$\\varepsilon>0$\u306b\u5bfe\u3057\u3066, $f \\in \\Sigma^{r,s}(\\Psi)$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\sum_{j=1}^s \\sup_{x \\in K} | f_j(x) - g_j(x) | < \\varepsilon\n  \\]\n  \u3068\u306a\u308b. \u3053\u3053\u3067, $f = (f_1,\\ldots,f_s)^{\\mathrm{T}}, g = (g_1,\\ldots,g_s)^{\\mathrm{T}}$\u3067\u3042\u308b.  \n\\end{thm}\n\\begin{proof}\n  \u5404$g_i \\in C(\\mathbb{R}^r,\\mathbb{R})$\u3092$\\Sigma^r(\\Psi)$\u306e\u5143\n  \\[\n  f_i = \\sum_{j=1}^{Q_i} \\beta_{i,j} \\Psi \\circ A_{i,j}\n  \\]\n  \u3067\u8fd1\u4f3c\u3059\u308b. \u3059\u308b\u3068, \n  \\[\n  f = \\sum_{i=1}^{s} \\sum_{j=1}^{Q_i} \\beta_{i,j} (\\Psi \\circ A_{i,j}) \\mathbf{e}_i\n  \\]\n  \u304c$g$\u306e\u8fd1\u4f3c\u3092\u4e0e\u3048\u308b.  \u3053\u3053\u306b$\\mathbf{e}_i$\u306f\u7b2c$i$\u6210\u5206\u304c$1$\u3067\u4ed6\u306e\u6210\u5206\u304c\u3059\u3079\u3066$0$\u3067\u3042\u308b$\\mathbb{R}^s$\u306e\u5143\u3067\u3042\u308b. \n\\end{proof}\n\n\u3055\u3066, \u3053\u308c\u307e\u3067\u8ff0\u3079\u3066\u304d\u305f\u7d50\u679c\u306f\u3059\u3079\u3066\u4e2d\u9593\u5c64\u304c$1$\u3064\u306e\u306e\u307f\u304b\u3089\u306a\u308b\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306b\u95a2\u3059\u308b\u3082\u306e\u3067\u3042\u3063\u305f. \u305d\u3053\u3067\u6b21\u306b, \u591a\u5c64\u304b\u3064\u591a\u51fa\u529b\u306e\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406\u306b\u3064\u3044\u3066\u8ff0\u3079\u3088\u3046. \u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u5c64\u6570\u3092$l \\geq 2$\u3067\u8868\u3059\u3053\u3068\u306b\u3059\u308b. \u305f\u3060\u3057, \u3053\u3053\u3067\u306f\u5165\u529b\u5c64\u306f\u5c64\u6570\u306b\u542b\u3081\u306a\u3044\u3053\u3068\u306b\u3059\u308b. \u3064\u307e\u308a, \u3053\u308c\u307e\u3067\u6271\u3063\u3066\u3044\u305f\u306e\u306f$l = 2$\u306e\u5834\u5408\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b. \n\n\\begin{defn}[$\\Sigma_l^{r,s}$\u30cd\u30c3\u30c8\u30ef\u30fc\u30af] \\\\\n  \u95a2\u6570$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306b\u5bfe\u3057\u3066, \u6b21\u306e\u3088\u3046\u306b\u96c6\u5408$I_1^{(q)},\\ldots,I_{l-1}^{(q)}, I_l ~(q \\in \\mathbb{N})$\u3092\u5e30\u7d0d\u7684\u306b\u5b9a\u7fa9\u3059\u308b. \n  \\[\n  \\begin{aligned}\n  I_1^{(q)} &= \\{ (\\Psi \\circ A_1, \\ldots , \\Psi \\circ A_q)^{\\mathrm{T}} \\mid A_1 \\ldots, A_q \\in \\mathbb{A}^r \\},  \\\\\n  I_{k+1}^{(q)} &= \\{ (\\Psi \\circ A_1, \\ldots, \\Psi \\circ A_q )^{\\mathrm{T}} \\circ f \\mid q' \\in \\mathbb{N} , f \\in I_{k}^{(q')}, A_j \\in \\mathbb{A}^{(q')} \\},  \\\\\n  I_l &= \\{  (A_1 , \\ldots, A_s )^{\\mathrm{T}} \\circ f \\mid q' \\in \\mathbb{N} , f \\in I_{l-1}^{(q')}, A \\in \\mathbb{A}^{(q')} \\}. \n  \\end{aligned}\n  \\]\n  \u3053\u306e\u3068\u304d, $I_l$\u3092$\\Psi$\u3092\u6d3b\u6027\u5316\u95a2\u6570\u3068\u3059\u308b$\\Sigma_l^{r,s}$\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3068\u547c\u3073, $\\Sigma_l^{r,s}(\\Psi)$\u3067\u8868\u3059. \n\\end{defn}\n\u3088\u308a\u5177\u4f53\u7684\u306b\u66f8\u3051\u3070$\\Sigma_l^{r,s}(\\Psi)$\u306e\u5143\u306f\u6b21\u306e\u3088\u3046\u306a$\\mathbb{R}^r$\u304b\u3089$\\mathbb{R}^s$\u3078\u306e\u95a2\u6570\u3067\u3042\u308b: \n$x \\in \\mathbb{R}^r$\u306b\u5bfe\u3057\u3066, $a_0 = x$\u3068\u304a\u304f. \n\u305d\u3057\u3066, $k = 1,\\ldots,l$\u306b\u5bfe\u3057\u3066, $q_0 = r, q_l = s, q_j \\in \\mathbb{N}$\u3068\u3057, $A_{k,i} \\in \\mathbb{A}^{q_{k-1}} ~~(i=1,\\ldots,q_k)$\u3068\u3059\u308b. \n\u307e\u305f, $G_1,\\ldots,G_{l-1} = \\Psi$\u3068\u3057, $G_l = \\mathrm{id}$\u3068\u3059\u308b. \n\u3053\u306e\u3068\u304d, \u6b21\u306e\u5f0f\n  \\[\n  a_{k,i} = G_k (A_{k,i}(a_{k-1})) ~~(i = 1,\\ldots,q_k, k = 1,\\ldots,l)\n  \\]\n\u306b\u3057\u305f\u304c\u3063\u3066\u8a08\u7b97\u3092\u7e70\u308a\u8fd4\u3057\u305f\u7d50\u679c$a_l=(a_{l,1},\\ldots,a_{l,s})^{\\mathrm{T}} \\in \\mathbb{R}^s$\u3092\u51fa\u529b\u3059\u308b. \n\n\u3053\u306e\u5b9a\u7fa9\u304b\u3089$\\Sigma^{r,s}(\\Psi) \\subset \\Sigma_2^{r,s}(\\Psi)$\u304c\u308f\u304b\u308b. \u5b9f\u969b, $f = \\sum_{j=1}^Q \\beta_j \\Psi \\circ A_j ~~(\\beta_j \\in \\mathbb{R}^s, A_j \\in A^r)$\u3068\u3059\u308b\u3068\u304d, \u5404$k = 1,\\ldots,Q$\u306b\u5bfe\u3057\u3066$B_k \\in \\mathbb{A}^Q$\u3092$B_k(x) = \\sum_{j=1}^Q \\beta_j^{(k)}x_j$\u3068\u5b9a\u7fa9\u3059\u308c\u3070, $f = (B_1,\\ldots,B_Q)^{\\mathrm{T}} \\circ (\\Psi \\circ A_1,\\ldots, \\Psi \\circ A_Q)^{\\mathrm{T}}$\u3068\u306a\u308b. \u3053\u306e\u3053\u3068\u304b\u3089, $\\Sigma^{r,s}(\\Psi)$\u304c\u4e07\u80fd\u8fd1\u4f3c\u80fd\u529b\u3092\u6301\u3064\u3068\u304d, $\\Sigma_2^{r,s}(\\Psi)$\u3082\u4e07\u80fd\u8fd1\u4f3c\u80fd\u529b\u3092\u6301\u3064\u3053\u3068\u304c\u308f\u304b\u308b. \u3088\u308a\u4e00\u822c\u306b\u6b21\u304c\u6210\u308a\u7acb\u3064. \n\\begin{thm}\\label{MainTheorem6}\n  \u95a2\u6570$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f$\\Sigma^r(\\Psi)$\u304c$C(\\mathbb{R}^r,\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u306b\u306a\u308b\u3082\u306e\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $\\Sigma_l^{r,s}(\\Psi)$\u306f$C(\\mathbb{R}^r,\\mathbb{R}^s)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{thm}\n\u3053\u306e\u5b9a\u7406\u3092\u793a\u3059\u305f\u3081\u306b\u3072\u3068\u3064\u88dc\u984c\u3092\u8a3c\u660e\u3057\u3088\u3046. \n\\begin{lem}\n  $\\mathcal{F}$\u306f\u300c$\\mathbb{R}$\u304b\u3089$\\mathbb{R}$\u3078\u306e\u95a2\u6570\u5168\u4f53\u306e\u96c6\u5408\u300d\u306e\u90e8\u5206\u96c6\u5408\u3067, $C(\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3068\u3057,  $\\mathcal{G}$\u306f\u300c$\\mathbb{R}^r$\u304b\u3089$\\mathbb{R}$\u3078\u306e\u95a2\u6570\u5168\u4f53\u306e\u96c6\u5408\u300d\u306e\u90e8\u5206\u96c6\u5408\u3067, $C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \n  \\[\n  F \\circ G = \\{f \\circ g \\mid f \\in \\mathcal{F}, g \\in \\mathcal{G}\\}\n  \\]\n  \u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{lem}\n\\begin{proof}\n  \u7a7a\u3067\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}^r$\u3092\u4efb\u610f\u306b\u53d6\u308b. $h \\in C(\\mathbb{R}^r)$\u3068\u3057, $\\varepsilon > 0$\u3092\u4efb\u610f\u306b\u53d6\u308b\u3068, \u4eee\u5b9a\u3088\u308a$g \\in \\mathcal{G}$\u304c\u5b58\u5728\u3057\u3066, $\\sup_{x \\in K} |g(x) - h(x)| < \\varepsilon$\u3068\u306a\u308b. \u3053\u306e\u3068\u304d, $h(K) \\subset \\mathbb{R}$\u306f\u30b3\u30f3\u30d1\u30af\u30c8, \u7279\u306b\u6709\u754c\u3067\u3042\u308b\u306e\u3067, $g(K)$\u3082\u6709\u754c\u3067\u3042\u308b. \u305d\u3053\u3067, $g(K)$\u306e\u9589\u5305$S := \\mathrm{cl}(g(K))$\u306f\u30b3\u30f3\u30d1\u30af\u30c8\u3067\u3042\u308b. \u3059\u308b\u3068, \u4eee\u5b9a\u3088\u308a\u9023\u7d9a\u95a2\u6570$\\mathbb{R} \\ni s \\mapsto s \\in \\mathbb{R}$\u306b\u5bfe\u3057\u3066, $f \\in C(\\mathbb{R})$\u304c\u5b58\u5728\u3057\u3066$\\sup_{s \\in S} |f(s) - s| < \\varepsilon$\u3068\u306a\u308b. \u3086\u3048\u306b, \u4efb\u610f\u306e$x \\in K$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  |f(g(x)) - h(x)| \n  &\\leq |f(g(x)) - g(x)| + |g(x) - h(x)| \\\\\n  &\\leq \\sup_{s \\in S} |f(s)-s| + \\sup_{z \\in K} |g(z)-h(z)| \\\\\n  &< \\varepsilon + \\varepsilon = 2\\varepsilon\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u3088\u3063\u3066, $\\sup_{x \\in K} |f(g(x)) - h(x)| \\leq 2\\varepsilon$\u3068\u306a\u308b. \n\\end{proof}\n\\newtheorem*{mainresult6}{\u5b9a\u7406\\ref{MainTheorem6}\u306e\u8a3c\u660e.}\n\\begin{mainresult6}\n  $s=1$\u306e\u5834\u5408\u3092\u8a3c\u660e\u3059\u308c\u3070\u5341\u5206\u3067\u3042\u308b. \n  \u6b21\u306e\u3088\u3046\u306b\u96c6\u5408$J_1,\\ldots,J_l$\u3092\u5e30\u7d0d\u7684\u306b\u5b9a\u7fa9\u3059\u308b. \n  \\[\n  \\begin{aligned}\n  J_1 &= \\{ \\sum_{j}^Q \\beta_j \\Psi \\circ A_j \\mid Q \\in \\mathbb{N}, A_j \\in \\mathbb{A}^r , \\beta_j \\in \\mathbb{R} \\},  \\\\\n  J_{k+1} &= \\{ \\sum_{j}^Q \\beta_j \\Psi \\circ A_j \\circ f \\mid Q \\in \\mathbb{N} , f \\in J_{k}, A_j \\in \\mathbb{A}^1 , \\beta_j \\in \\mathbb{R} \\},  \\\\\n  \\end{aligned}\n  \\]\n  \u3053\u306e\u3068\u304d, $x \\mapsto \\sum_{j}\\beta_j x_j$\u306f\u7dda\u5f62\u5199\u50cf\u306a\u306e\u3067$J_l \\subset \\Sigma_{l+1}^{r,s}(\\Psi)$\u3067\u3042\u308b. \n  \u305d\u3053\u3067, $J_l$\u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044. \u3053\u306e\u3053\u3068\u3092$l$\u306b\u95a2\u3059\u308b\u5e30\u7d0d\u6cd5\u3067\u793a\u305d\u3046. \u307e\u305a$\\Sigma^r(\\Psi) = J_1$\u3067\u3042\u308b\u306e\u3067$J_1$\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \u6b21\u306b, $J_k $\u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3068\u3059\u308b\u3068, $J_{k+1} = \\Sigma^1(\\Psi) \\circ J_k$\u3067\u3042\u308a, $\\Sigma^1(\\Psi)$\u306f$C(\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u306e\u3067, \u524d\u88dc\u984c\u3088\u308a$J_{k+1}$\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \\qed\n\\end{mainresult6}\n\n\\subsection{$L^p$\u7a7a\u9593\u306b\u304a\u3051\u308b\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406}\n\u672c\u7bc0\u3067\u306f, \u524d\u7ae0\u306e\u7d50\u679c\u3092\u4f7f\u3063\u3066$\\mathbb{R}^r$\u4e0a\u306eBorel\u53ef\u6e2c\u95a2\u6570\u5168\u4f53\u306e\u7a7a\u9593$\\mathbb{M}^r := \\mathbb{M}(\\mathbb{R}^r,\\mathbb{R})$\u3084$L^p$\u7a7a\u9593\u306b\u304a\u3051\u308b$\\Sigma^r(\\Psi)$\u306e\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406\u3092\u8ff0\u3079\u308b. \u307e\u305a, $C(\\mathbb{R}^r)$\u304c$\\mathbb{M}^r$\u306b\u304a\u3044\u3066\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u306b\u3064\u3044\u3066\u8ff0\u3079\u308b. \n\\begin{defn}\n  $\\mu$\u3092$(\\mathbb{R}^r,\\mathcal{B}^r)$\u4e0a\u306e\u6709\u9650\u6e2c\u5ea6\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $f,g \\in \\mathbb{M}^r$\u304c$\\mu$-\u540c\u5024\u3067\u3042\u308b\u3068\u306f, $\\mu(\\{f \\neq g\\}) = 0$\u3068\u306a\u308b\u3053\u3068\u3092\u3044\u3046. \u4ee5\u5f8c, \u3053\u306e\u540c\u5024\u95a2\u4fc2\u306b\u3088\u308b$\\mathbb{M}^r$\u306e\u5546\u96c6\u5408\u3092\u305d\u306e\u307e\u307e$\\mathbb{M}^r$\u3067\u8868\u3059\u5834\u5408\u304c\u3042\u308b. \n\\end{defn}\n\\begin{defn}\n  $\\mu$\u3092$(\\mathbb{R}^r,\\mathcal{B}^r)$\u4e0a\u306e\u6709\u9650\u6e2c\u5ea6\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $\\rho_{\\mu}:\\mathbb{M}^r \\times \\mathbb{M}^r \\rightarrow [0,\\infty)$\u3092, \n  \\[\n  \\rho_{\\mu}(f,g) := \\inf \\{ \\varepsilon > 0 \\mid \\mu({|f-g|>\\varepsilon}) < \\varepsilon \\}\n  \\]\n  \u306b\u3088\u308a\u5b9a\u7fa9\u3059\u308b(\u6709\u9650\u6e2c\u5ea6\u306a\u306e\u3067$\\rho_{\\mu}(f,g) \\leq \\mu(\\mathbb{R}^r) < \\infty$\u3068\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f). \n\\end{defn}\n\\begin{lem}\\label{ProbabilityConvergenceEq}\n  $\\mu$\u3092$(\\mathbb{R}^r,\\mathcal{B}^r)$\u4e0a\u306e\u6709\u9650\u6e2c\u5ea6\u3068\u3059\u308b. \u307e\u305f, $(f_n)_{n=1}^{\\infty}$\u3092$\\mathbb{M}^r$\u306e\u5143\u306e\u5217\u3068\u3057, $f \\in \\mathbb{M}^r$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d\n  , \u6b21\u306e\u6761\u4ef6\u306f\u540c\u5024\u3067\u3042\u308b. \n  \\begin{itemize}\n      \\item[(1):]$\\rho_{\\mu}(f_n,f) \\rightarrow 0.$ \n      \\item[(2):]$\\forall \\varepsilon > 0 , ~\\mu(\\{|f_n - f| > \\varepsilon\\}) \\rightarrow 0.$ \n      \\item[(3):]$\\int_{\\mathbb{R}^r} \\min(|f_n-f|, 1) d\\mu \\rightarrow 0.$\n  \\end{itemize}\n\\end{lem}\n\\begin{proof} \\\\\n  $(1) \\Rightarrow (2)$: \u4efb\u610f\u306b$\\varepsilon > 0$\u3092\u53d6\u308b. \n  \u3044\u307e, \u4eee\u5b9a\u3088\u308a\u3042\u308b$N \\in \\mathbb{N}$\u304c\u5b58\u5728\u3057\u3066\u4efb\u610f\u306e$n \\geq N$\u306b\u5bfe\u3057\u3066$\\rho_{\\mu}(f_n,f) < \\varepsilon$\u3068\u306a\u308b. \n  \u3059\u308b\u3068, \u4efb\u610f\u306e$n \\geq N$\u306b\u5bfe\u3057\u3066, $\\rho_{\\mu}$\u306e\u5b9a\u7fa9\u3088\u308a$\\mu(|f_n - f| > \\varepsilon_n) < \\varepsilon_n$\u306a\u308b$0 < \\varepsilon_n < \\varepsilon$\u304c\u53d6\u308c\u308b. \n  \u3086\u3048\u306b, \u4efb\u610f\u306e$n \\geq N$\u306b\u5bfe\u3057\u3066, $\\mu(|f_n - f| > \\varepsilon) \\leq \\mu(|f_n - f| > \\varepsilon_n) < \\varepsilon_n < \\varepsilon$\u3068\u306a\u308b.\n  \u3057\u305f\u304c\u3063\u3066, \u4efb\u610f\u306b$\\delta > 0$\u3092\u53d6\u308b\u3068, \u3042\u308b$M \\in \\mathbb{N}$\u304c\u5b58\u5728\u3057\u3066\u4efb\u610f\u306e$n \\geq M$\u306b\u5bfe\u3057\u3066$\\mu(|f_n - f| > \\delta) < \\delta$\u3068\u306a\u308b. \u305d\u3053\u3067, $\\varepsilon \\leq \\delta$\u306e\u5834\u5408\u306f, \u4efb\u610f\u306e$n \\geq N$\u306b\u5bfe\u3057\u3066, $\\mu(|f_n - f| > \\varepsilon) < \\varepsilon \\leq \\delta$\u3068\u306a\u308b. $\\varepsilon > \\delta$\u306e\u5834\u5408\u306f, \u4efb\u610f\u306e$n \\geq M$\u306b\u5bfe\u3057\u3066$\\mu(|f_n - f| > \\varepsilon) \\leq \\mu(|f_n - f| > \\delta) < \\delta$\u3068\u306a\u308b. \u3053\u308c\u306f$\\mu(\\{|f_n - f| > \\varepsilon\\}) \\rightarrow 0$\u3092\u610f\u5473\u3059\u308b. \\\\\n  $(2) \\Rightarrow (1)$: \u4efb\u610f\u306b$\\varepsilon > 0$\u3092\u53d6\u308b. \u3044\u307e, \u4eee\u5b9a\u3088\u308a\u3042\u308b$N \\in \\mathbb{N}$\u304c\u5b58\u5728\u3057\u3066\u4efb\u610f\u306e$n \\geq N$\u306b\u5bfe\u3057\u3066$\\mu(|f_n - f| > \\varepsilon) < \\varepsilon$\u3068\u306a\u308b. \u3086\u3048\u306b, \u4efb\u610f\u306e$n \\geq N$\u306b\u5bfe\u3057\u3066$\\rho_{\\mu}(f_n,f) \\leq \\varepsilon$\u3068\u306a\u308b. \\\\\n  $(2) \\Rightarrow (3)$: $\\varepsilon > 0$\u3092\u4efb\u610f\u306b\u53d6\u308b. \u3053\u306e\u3068\u304d, \u4eee\u5b9a\u3088\u308a\u3042\u308b$N \\in \\mathbb{N}$\u304c\u5b58\u5728\u3057\u3066\u4efb\u610f\u306e$n \\geq N$\u306b\u5bfe\u3057\u3066$\\mu(|f_n - f| > \\varepsilon) < \\varepsilon$\u3068\u306a\u308b. \u305d\u3053\u3067, \u4efb\u610f\u306e$n \\geq N$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  \\int_{\\mathbb{R}^r} \\min(|f_n - f|,1) d\\mu\n  &\\leq \\int_{|f_n-f|>\\varepsilon} 1 d\\mu + \\int_{|f_n-f|\\leq \\varepsilon} |f_n - f| d\\mu \\\\\n  &\\leq \\varepsilon + \\varepsilon \\mu(\\mathbb{R}^r)\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. $\\mu$\u306f\u6709\u9650\u6e2c\u5ea6\u306a\u306e\u3067\u3053\u308c\u3067$(3)$\u306e\u6210\u7acb\u304c\u78ba\u304b\u3081\u3089\u308c\u305f. \\\\\n  $(3) \\Rightarrow (2)$: \u4efb\u610f\u306b$1 > \\varepsilon > 0$\u3092\u53d6\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$\\delta > 0$\u306b\u5bfe\u3057\u3066, \u3042\u308b$N \\in \\mathbb{N}$\u304c\u5b58\u5728\u3057\u3066\u4efb\u610f\u306e$n \\geq N$\u306b\u5bfe\u3057\u3066, $\\int_{\\mathbb{R}^r} \\min(|f_n-f|,1)d\\mu < \\varepsilon \\delta$\u3068\u306a\u308b. \u305d\u3053\u3067, \u4efb\u610f\u306e$n \\geq N$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  \\mu(|f_n - f| > \\varepsilon)\n  &\\leq \\frac{1}{\\varepsilon} \\int_{|f_n - f| > \\varepsilon} \\min(|f_n-f|,1) d\\mu \\\\\n  &\\leq \\frac{1}{\\varepsilon} \\int_{\\mathbb{R}^r} \\min(|f_n-f|,1) d\\mu \\\\\n  &\\leq \\delta\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, $\\mu(\\{|f_n - f| > \\varepsilon\\}) \\rightarrow 0$\u3067\u3042\u308b. $\\varepsilon \\geq 1$\u306e\u5834\u5408\u3082$0 < \\varepsilon' < 1$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\mu(|f_n - f| > \\varepsilon) \\leq \\mu(|f_n - f| > \\varepsilon') \\rightarrow 0.\n  \\]\n  \u3068\u306a\u308b\u306e\u3067\u6210\u7acb\u3059\u308b. \n\\end{proof}\n\\begin{lem}\\label{DensityOfCinM}\n  $\\mu$\u3092$(\\mathbb{R}^r,\\mathcal{B}^r)$\u4e0a\u306e\u6709\u9650\u6e2c\u5ea6\u3068\u3059\u308b\u3068\u304d, $C(\\mathbb{R}^r)$\u306f$\\mathbb{M}^r$\u306b\u304a\u3044\u3066$\\rho_{\\mu}$-\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{lem}\n\\begin{proof}\n  $f \\in \\mathbb{M}^r$\u3068\u3057, $\\varepsilon > 0$\u3092\u4efb\u610f\u306b\u53d6\u308b. \u512a\u53ce\u675f\u5b9a\u7406\u306b\u3088\u308a, \u5341\u5206\u5927\u304d\u306a$M \\geq 0$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\int_{\\mathbb{R}^r} \\min(|f\\cdot\\chi_{\\{|f| \\geq M \\}} - f|, 1)d\\mu < \\varepsilon\n  \\]\n  \u304c\u6210\u7acb\u3059\u308b. \u3053\u3053\u3067, $f\\cdot\\chi_{\\{|f| \\geq M \\}} \\in L^1(\\mathbb{R}^r,\\mathcal{B}^r,\\mu)$\u3067\u3042\u308b\u304b\u3089, $C_0(\\mathbb{R}^r)$\u304c$L^1(\\mathbb{R}^r,\\mathcal{B}^r,\\mu)$\u306b\u304a\u3044\u3066$L^1$\u30ce\u30eb\u30e0\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\n  (\u4ed8\u9332\u306e\u88dc\u984c\\ref{DensityOfC0InLp})\u3088\u308a, \u3042\u308b$g \\in C(\\mathbb{R}^r)$\u3067$\\int_{\\mathbb{R}^r} |f\\cdot\\chi_{\\{|f| \\geq M \\}} - g| d\\mu < \\varepsilon$\u3068\u306a\u308b. \u3088\u3063\u3066, \n  \\[\n  \\begin{aligned}\n  \\int_{\\mathbb{R}^r} \\min(|f-g|,1) d\\mu\n  &\\leq \\int_{R^r} \\min(|f - f\\cdot\\chi_{\\{|f| \\geq M \\}}|,1) d\\mu \\\\\n  &~~~ + \\int_{R^r} \\min(|f\\cdot\\chi_{\\{|f| \\geq M \\}} - g|,1) d\\mu \\\\\n  &\\leq \\varepsilon + \\int_{R^r} |f\\cdot\\chi_{\\{|f| \\geq M \\}} - g| d\\mu \\\\\n  &\\leq \\varepsilon + \\varepsilon = 2\\varepsilon\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3053\u308c\u3088\u308a\u4efb\u610f\u306e$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066$g \\in C(\\mathbb{R}^r)$\u304c\u5b58\u5728\u3057\u3066$\\rho_{\\mu}(f,g) < \\varepsilon$\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b(\u88dc\u984c\\ref{ProbabilityConvergenceEq}\u306e$(3) \\Rightarrow (2)$\u306e\u8a3c\u660e\u53c2\u7167).\n\\end{proof}\n\u6b21\u306b$\\Sigma^r(\\Psi)$\u304c$\\mathbb{M}^r$\u306b\u304a\u3044\u3066\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u306e\u8a3c\u660e\u306b\u4f7f\u3046\u88dc\u984c\u3092\u793a\u3059. \n\\begin{lem}\\label{CompactUniConverThenProbConver}\n  $(f_n)_{n=1}^{\\infty}$\u3092$\\mathbb{M}^r$\u306e\u5143\u306e\u5217\u3068\u3057, $f \\in \\mathbb{M}^r$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $(f_n)_{n=1}^{\\infty}$\u304c$f$\u306b\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u3059\u308b\u306a\u3089\u3070, \u4efb\u610f\u306e$(\\mathbb{R}^r,\\mathcal{B}^r)$\u4e0a\u306e\u6709\u9650\u6e2c\u5ea6$\\mu$\u306b\u5bfe\u3057\u3066, $\\rho_{\\mu}(f_n,f) \\rightarrow 0$\u3068\u306a\u308b. \n\\end{lem}\n\\begin{proof}\n  \u4efb\u610f\u306b$\\varepsilon > 0$\u3092\u53d6\u308b. \n  $\\mu$\u306f\u6709\u9650\u6e2c\u5ea6\u306a\u306e\u3067\u4ed8\u9332\u306e\u7cfb\\ref{FiniteBorelMeasOnRnIsRegular}\u3088\u308a\u6b63\u5247\u3067\u3042\u308b.\n  \u3057\u305f\u304c\u3063\u3066, \u3042\u308b\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\in \\mathbb{R}^r$\u304c\u5b58\u5728\u3057\u3066$\\mu(\\mathbb{R}^r \\setminus K) < \\varepsilon$\u3068\u306a\u308b. \n  \u305d\u3053\u3067\u4eee\u5b9a\u3088\u308a, \u3042\u308b$N \\in \\mathbb{N}$\u304c\u5b58\u5728\u3057\u3066, \u4efb\u610f\u306e$n \\geq N$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\sup_{x \\in K} |f_n(x) - f(x)| < \\varepsilon\n  \\]\n  \u3068\u306a\u308b. \u3086\u3048\u306b, \u4efb\u610f\u306e$n \\geq N$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  \\int_{\\mathbb{R}^r} \\min(|f_n-f|,1)d\\mu \n  &= \\int_{K} \\min(|f_n-f|,1)d\\mu + \\int_{\\mathbb{R}^r \\setminus K} \\min(|f_n-f|,1)d\\mu \\\\\n  &\\leq \\int_{K} |f_n-f| d\\mu + \\int_{\\mathbb{R}^r \\setminus K} 1 d\\mu \\\\\n  &\\leq \\varepsilon \\mu(\\mathbb{R}^r) + \\varepsilon \\\\\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, $\\int_{\\mathbb{R}^r} \\min(|f_n-f|,1)d\\mu  \\rightarrow 0$\u306a\u306e\u3067, \u88dc\u984c\\ref{ProbabilityConvergenceEq}\u3088\u308a$\\rho_{\\mu}(f_n,f) \\rightarrow 0$\u3067\u3042\u308b. \n\\end{proof}\n\n\\begin{thm}\\label{UATinMr}\n  \u95a2\u6570$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f$\\Sigma^r(\\Psi)$\u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u306b\u306a\u308b\u3082\u306e\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$(\\mathbb{R}^r,\\mathcal{B}^r)$\u4e0a\u306e\u6709\u9650\u6e2c\u5ea6$\\mu$\u306b\u5bfe\u3057\u3066, $\\Sigma^r(\\Psi)$\u306f$\\mathbb{M}^r$\u306b\u304a\u3044\u3066$\\rho_{\\mu}$-\u7a20\u5bc6\u3067\u3042\u308b.\n\\end{thm}\n\\begin{proof}\n  $\\Sigma^r(\\Psi)$\u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3068\u88dc\u984c\\ref{CompactUniConverThenProbConver}\u3088\u308a$\\Sigma^r(\\Psi)$\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066$\\rho_{\\mu}$-\u7a20\u5bc6\u3067\u3042\u308b. \u3053\u306e\u3053\u3068\u3068 \u88dc\u984c\\ref{DensityOfCinM}\u3092\u5408\u308f\u305b\u308c\u3070$\\Sigma^r(\\Psi)$\u306f$\\mathbb{M}^r$\u306b\u304a\u3044\u3066$\\rho_{\\mu}$-\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \n\\end{proof}\n\u6b21\u306e\u5b9a\u7406\u306f$\\mathbb{M}^r$\u3092sup\u30ce\u30eb\u30e0\u306e\u610f\u5473\u3067$\\Sigma^r(\\Psi)$\u306b\u3088\u308a\u5341\u5206\u3088\u304f\u8fd1\u4f3c\u3067\u304d\u308b\u3053\u3068\u3092\u793a\u3059\u3082\u306e\u3067\u3042\u308b. \n\\begin{thm}\\label{DensityOfNNinMr}\n  \u95a2\u6570$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f$\\Sigma^r(\\Psi)$\u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u306b\u306a\u308b\u3082\u306e\u3068\u3059\u308b.  \u307e\u305f\n  $\\mu$\u3092$(\\mathbb{R}^r,\\mathcal{B}^r)$\u4e0a\u306e\u6709\u9650\u6e2c\u5ea6\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$g \\in \\mathbb{M}^r$\u3068$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, $f \\in \\Sigma^r(\\Psi)$\u3068\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}^r$\u304c\u5b58\u5728\u3057\u3066, $\\mu(K) > \\mu(\\mathbb{R}^r) - \\varepsilon$\u304b\u3064$\\forall x \\in K , |f(x) - g(x)| < \\varepsilon$\u3068\u306a\u308b. \n\\end{thm}\n\u8a3c\u660e\u306b\u306f\u6b21\u306e\u4e8b\u5b9f\u3092\u4f7f\u3046(\u5bae\u5cf6\\cite{miyajima}\u306e\u7b2c7\u7ae02\u7bc0\u306a\u3069\u3092\u53c2\u7167\u3055\u308c\u305f\u3044). \n\\begin{thm}[Tietze\u306e\u62e1\u5f35\u5b9a\u7406]\\label{TietzeTheorem} \\\\\n  $X$\u306fHausdorff\u7a7a\u9593\u3067, \u4efb\u610f\u306e\u4ea4\u308f\u3089\u306a\u3044\u9589\u96c6\u5408$F_1,F_2 \\subset X$\u306b\u5bfe\u3057\u3066$F_1 \\subset U_1, F_2 \\subset U_2$\u304b\u3064$U_1 \\cap U_2 = \\emptyset$\u3068\u306a\u308b\u958b\u96c6\u5408$U_1,U_2 \\subset X$\u304c\u5b58\u5728\u3059\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u9589\u96c6\u5408$A \\subset X$\u4e0a\u306e\u4efb\u610f\u306e\u5b9f\u6570\u5024\u9023\u7d9a\u95a2\u6570$f$\u306b\u5bfe\u3057\u3066$X$\u4e0a\u306e\u5b9f\u6570\u5024\u9023\u7d9a\u95a2\u6570$f'$\u304c\u5b58\u5728\u3057\u3066$f = f'|_A$\u304c\u6210\u7acb\u3059\u308b. \u3064\u307e\u308a, $f$\u3092$X$\u4e0a\u306e\u5b9f\u6570\u5024\u9023\u7d9a\u95a2\u6570\u306b\u62e1\u5f35\u3067\u304d\u308b. \n\\end{thm}\n\\renewcommand{\\proofname}{\\bf \u5b9a\u7406\\ref{DensityOfNNinMr}\u306e\u8a3c\u660e.}\n\\begin{proof}\n  Lusin\u306e\u5b9a\u7406(\u4ed8\u9332\u306e\u547d\u984c\\ref{LusinTheorem})\u3088\u308a, \u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}^r$\u304c\u5b58\u5728\u3057\u3066, $\\mu(K) > \\mu(\\mathbb{R}^r) - \\varepsilon$\u304b\u3064$K$\u4e0a\u306e\u95a2\u6570$g|K$\u306f\u9023\u7d9a\u3068\u306a\u308b.\n  \u305d\u3053\u3067Tietze\u306e\u62e1\u5f35\u5b9a\u7406\u3088\u308a, $g' \\in C(\\mathbb{R}^r)$\u304c\u5b58\u5728\u3057\u3066$g|K = g'|K$\u3068\u306a\u308b. \u3044\u307e $\\Sigma^r(\\Psi)$\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u306e\u3067, $f \\in \\Sigma^r(\\Psi)$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\sup_{x \\in K} |f(x) - g(x)| = \\sup_{x \\in K} |f(x) - g'(x)| < \\varepsilon. \n  \\]\n\\end{proof}\n\\renewcommand{\\proofname}{\\bf \u8a3c\u660e.}\n\u3055\u3066, $L^p$\u7a7a\u9593\u306b\u304a\u3051\u308b\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406\u306b\u8a71\u3092\u79fb\u305d\u3046. \n\\begin{defn}\n  $f,g \\in L^p(\\mathbb{R},\\mathcal{B}^r,\\mu)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  d_{\\mu}^p(f,g) = \\left( \\int_{\\mathbb{R}^r} |f-g|^p d\\mu \\right)^{1\/p}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. $d_{\\mu}^p$\u306f$L^p(\\mathbb{R}^r,\\mathcal{B}^r,\\mu)$\u4e0a\u306e\u8ddd\u96e2\u3067\u3042\u308b. \n\\end{defn}\n\\begin{thm}\n  \u95a2\u6570$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f$\\Sigma^r(\\Psi)$\u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u306b\u306a\u308b\u3082\u306e\u3068\u3059\u308b. \u307e\u305f$\\mu$\u3092$(\\mathbb{R}^r,\\mathcal{B}^r)$\u4e0a\u306e\u6709\u9650\u6e2c\u5ea6\u3068\u3059\u308b. \n  \u3053\u306e\u3068\u304d, $\\mu(K)=\\mu(\\mathbb{R}^r)$\u306a\u308b\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}^r$\u304c\u5b58\u5728\u3059\u308c\u3070, \u4efb\u610f\u306e$p \\in [1,\\infty)$\u306b\u3064\u3044\u3066,  $\\Sigma^r(\\Psi)$\u306f$L^p(\\mathbb{R}^r,\\mathcal{B}^r,\\mu)$\u306b\u304a\u3044\u3066$d_{\\mu}^p$-\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{thm}\n\\begin{proof}\n  $g \\in L^p(\\mathbb{R}^r,\\mathcal{B}^r,\\mu)$\u3068$\\varepsilon > 0$\u3092\u4efb\u610f\u306b\u53d6\u308b. \n  \u4ed8\u9332\u306e\u88dc\u984c\\ref{DensityOfC0InLp}\u3088\u308a $C_0(\\mathbb{R}^r)$\u306f\\\\\n  $L^p(\\mathbb{R}^r,\\mathcal{B}^r,\\mu)$\u306b\u304a\u3044\u3066$d_{\\mu}^p$-\u7a20\u5bc6\u3067\u3042\u308b\u306e\u3067, $h \\in C(\\mathbb{R}^r)$\u3067$d_{\\mu}^p(g,h) < \\varepsilon$\u306a\u308b\u3082\u306e\u304c\u53d6\u308c\u308b. \u3044\u307e, $\\Sigma^r(\\Psi)$\u306f$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u306e\u3067, $f \\in \\Sigma^r(\\Psi)$\u304c\u5b58\u5728\u3057\u3066$\\sup_{x \\in K} |h(x) - f(x)| < \\varepsilon$\u3068\u306a\u308b. \u3088\u3063\u3066, $\\mu(K) = \\mu(\\mathbb{R}^r)$\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u3068, \n  \\[\n  \\begin{aligned}\n  d_{\\mu}^p(g,f) \\leq d_{\\mu}^p(g,h) + d_{\\mu}^p(h,f) \n  \\leq \\varepsilon + (\\int_{K} \\varepsilon^p d\\mu)^{1\/p} \\leq \\varepsilon + \\varepsilon (\\mu(\\mathbb{R}^r))^{1\/p}\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \n\\end{proof}\n\\begin{cor}\n  \u95a2\u6570$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f$\\Sigma^r(\\Psi)$\u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u306b\u306a\u308b\u3082\u306e\u3068\u3059\u308b. \u307e\u305f, $K \\subset \\mathbb{R}^r$\u3092\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u3068\u3057$\\mu$\u3092$(K,\\mathcal{B}_{K})$\u4e0a\u306e\u6709\u9650\u6e2c\u5ea6\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$p \\in [1,\\infty)$\u306b\u3064\u3044\u3066,  $\\Sigma^r(\\Psi)$(\u3092$K$\u306b\u5236\u9650\u3057\u305f\u3082\u306e)\u306f$L^p(K,\\mathcal{B}_{K},\\mu)$\u306b\u304a\u3044\u3066$d_{\\mu}^p$-\u7a20\u5bc6\u3067\u3042\u308b. \u305f\u3060\u3057, $\\mathcal{B}_{K}$\u306f$K$\u4e0a\u306eBorel $\\sigma$-\u52a0\u6cd5\u65cf\u3067\u3042\u308a, $\\mathcal{B}_{K} = \\{K \\cap A \\mid A \\in \\mathcal{B}^r\\}$\u3092\u6e80\u305f\u3059. \n\\end{cor}\n\\begin{proof}\n  $(\\mathbb{R}^r,\\mathcal{B}^r)$\u4e0a\u306e\u6709\u9650\u6e2c\u5ea6$\\tilde{\\mu}$\u3092$\\tilde{\\mu}(A) = \\mu(A \\cap K)$\u3067\u5b9a\u3081, $f \\in L^p(K,\\mathcal{B}_{K},\\mu)$\u306b\u5bfe\u3057$\\tilde{f} \\in L^o(\\mathbb{R}^r,\\mathcal{B}^r,\\tilde{\\mu})$\u3092\n  \\[\n  \\tilde{f}(x) = \n  \\begin{cases}\n  f(x) ~~~(x \\in K)\\\\\n  0 ~~~~~~~~(\\mathrm{otherwise})\n  \\end{cases}\n  \\]\n  \u3067\u5b9a\u3081\u308b. \u3053\u306e\u3068\u304d, \u3059\u3050\u4e0a\u306e\u5b9a\u7406\u3088\u308a, \u4efb\u610f\u306e$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066. $g \\in \\Sigma^r(\\Psi)$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  \\varepsilon > d_{\\tilde{\\mu}}^p(\\tilde{f},g) = \\left( \\int_{\\mathbb{R}^r} |\\tilde{f}-g|^p d_{\\tilde{\\mu}} \\right)^{1\/p} = \\left( \\int_{K} |f-g|^p d\\mu  \\right)^{1\/p} = d_{\\mu}^p(f,g|_{K})\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \n\\end{proof}\n\n\u4ee5\u4e0b\u306fLeshno et al.\u306e\u7d50\u679c\u306e\u7cfb\u3067\u3042\u308b. \n\\begin{cor}\n  $\\mu$\u3092$(\\mathbb{R}^r,\\mathcal{B}_{\\mathbb{R}^r})$\u4e0a\u306e\u6709\u9650\u6e2c\u5ea6\u3067, $\\mu(K)=\\mu(\\mathbb{R}^r)$\u306a\u308b\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}^r$\u304c\u5b58\u5728\u3059\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $\\Psi \\in \\mathcal{M}(\\mathbb{R})$\u306b\u5bfe\u3057\u3066,  $\\Psi$\u304c\u3069\u3093\u306a\u591a\u9805\u5f0f\u95a2\u6570\u3068\u3082a.e.\u3067\u4e00\u81f4\u3059\u308b\u3053\u3068\u304c\u306a\u3044\u3053\u3068\u3068 $\\Sigma^r(\\Psi)$\u304c$L^p(\\mathbb{R}^r,\\mathcal{B}_{\\mathbb{R}^r},\\mu) ~(p \\geq 1)$\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u306f\u540c\u5024\u3067\u3042\u308b. \n\\end{cor}\n\\begin{proof}\n  $\\Psi$\u304c\u3042\u308b$m$\u6b21\u591a\u9805\u5f0f\u95a2\u6570\u3068$a.e.$\u3067\u4e00\u81f4\u3059\u308b\u3068\u3059\u308b\u3068, $\\Sigma^r(\\Psi)$\u306f$r$\u5909\u6570\u306e$m$\u6b21\u4ee5\u4e0b\u306e\u591a\u9805\u5f0f\u95a2\u6570\u3068a.e.\u3067\u4e00\u81f4\u3059\u308b\u95a2\u6570\u5168\u4f53\u306e\u96c6\u5408\u306b\u306a\u308b\u306e\u3067$L^p(\\mathbb{R}^r,\\mathcal{B}_{\\mathbb{R}^r},\\mu)$\u306b\u304a\u3044\u3066$m+1$\u6b21\u306e\u591a\u9805\u5f0f\u95a2\u6570\u3092\u8fd1\u4f3c\u3059\u308b\u3053\u3068\u306f\u3067\u304d\u306a\u3044($\\mu(K)=\\mu(\\mathbb{R}^r)$\u3086\u3048\u591a\u9805\u5f0f\u95a2\u6570\u306f$L^p(\\mu)$\u306b\u5c5e\u3059\u308b\u3053\u3068\u306b\u6ce8\u610f). \u9006\u306b$\\Psi$\u304c\u3069\u3093\u306a\u591a\u9805\u5f0f\u95a2\u6570\u3068\u3082a.e.\u3067\u4e00\u81f4\u3059\u308b\u3053\u3068\u304c\u306a\u3044\u3068\u3057, $f \\in L^p(\\mathbb{R}^r,\\mathcal{B}_{\\mathbb{R}^r},\\mu)$\u3092\u3068\u308b. \u3044\u307e,  \u4ed8\u9332\u306e\u547d\u984c\\ref{DensityOfC0InLp}\u3088\u308a$C(K)$\u306f$L^p(K,\\mu)$\u3067\u7a20\u5bc6\u3067\u3042\u308b\u306e\u3067, $g \\in C(K)$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\lVert f-g \\rVert_{L^p}^p = \\int_{\\mathbb{R}^r} |f-g|^p d\\mu = \\int_{K} |f-g| d\\mu < (\\varepsilon\/2)^p\n  \\]\n  \u3068\u306a\u308b. \u5b9a\u7406\\ref{LeshnoMainResult1}\u3088\u308a$\\Sigma^r(\\Psi)$\u306f$C(K)$\u306b\u304a\u3044\u3066\u7a20\u5bc6\u3067\u3042\u308b\u306e\u3067, $h \\in \\Sigma^r(\\Psi))$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\sup_{x \\in K} |g(x)-h(x)| < \\varepsilon\/{\\mu(K)^{1\/p}}\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, \n  \\[\n  \\lVert f - h \\rVert_{L^p} \\leq \\lVert f - g \\rVert_{L^p} + \\lVert g - h \\rVert_{L^p} \\leq \\varepsilon.\n  \\]\n\\end{proof}\n\u4ee5\u4e0a\u306e\u7d50\u679c\u306f\u591a\u5c64\u30fb\u591a\u51fa\u529b\u306e\u5834\u5408\u3082\u6210\u7acb\u3059\u308b. \u8a3c\u660e\u306f$C(\\mathbb{R}^r,\\mathbb{R}^s)$\u306e\u5834\u5408\u3068\u540c\u69d8\u306b\u3067\u304d\u308b. \n\\begin{defn}\n  \u6b21\u306e\u3088\u3046\u306b\u5b9a\u3081\u308b. \n  \\[\n  \\begin{aligned}\n  &M^{r,s} = M(\\mathbb{R}^r,\\mathbb{R}^s) := \\{f:\\mathbb{R}^r \\rightarrow \\mathbb{R}^s \\mid f\\mbox{\u306f}(\\mathcal{B}^r,\\mathcal{B}^s)\\mbox{\u53ef\u6e2c}\\} \\\\\n  &L^p(r,s,\\mu) := \\{ f = (f_1,\\ldots,f_s) \\in M^{r,s} \\mid \\int_{\\mathbb{R}^r} |f_j|^p d\\mu < \\infty ~~(j=1,\\ldots,s) \\}\n  \\end{aligned}\n  \\]\n\\end{defn}\n\\begin{thm}\n  \u95a2\u6570$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f$\\Sigma^r(\\Psi)$\u304c$C(\\mathbb{R}^r,\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u306b\u306a\u308b\u3082\u306e\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $(\\mathbb{R}^r,\\mathcal{B}^r)$\u4e0a\u306e\u6709\u9650\u6e2c\u5ea6$\\mu$\u306b\u5bfe\u3057\u3066, $\\Sigma^{r,s}(\\Psi)$\u306f$\\mathbb{M}^{r,s} := \\mathbb{M}(\\mathbb{R}^r,\\mathbb{R}^s)$\u306b\u304a\u3044\u3066$\\rho_{\\mu}^s$-\u7a20\u5bc6\u3067\u3042\u308b. \n  \u305f\u3060\u3057, $f,g \\in \\mathbb{M}^{r,s}$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\rho_{\\mu}^s(f,g) = \\sum_{j=1}^s \\inf \\left\\{ \\varepsilon>0 ~\\vline~ \\mu(|f_j-g_j| > \\varepsilon ) < \\varepsilon \\right\\}\n  \\]\n  \u3067\u3042\u308b. \u3055\u3089\u306b, $\\mu(K) = \\mu(\\mathbb{R}^r)$\u306a\u308b\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}^r$\u304c\u5b58\u5728\u3059\u308c\u3070, \u4efb\u610f\u306e$p \\geq 1$\u306b\u5bfe\u3057\u3066, $\\Sigma^{r,s}(\\Psi)$\u306f$L^p(r,s,\\mu)$\u306b\u304a\u3044\u3066$d_{\\mu}^{p,s}$-\u7a20\u5bc6\u3067\u3042\u308b. \u305f\u3060\u3057, $f,g \\in L^p(r,s,\\mu)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  d_{\\mu}^{p,s}(f,g) = \\sum_{j=1}^s \\left( \\int_{\\mathbb{R}^r} |f_j-g_j|^p d\\mu \\right)^{1\/p}\n  \\]\n  \u3067\u3042\u308b. \n\\end{thm}\n\\begin{thm}\n  \u95a2\u6570$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f$\\Sigma^r(\\Psi)$\u304c$C(\\mathbb{R}^r,\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u306b\u306a\u308b\u3082\u306e\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$(\\mathbb{R}^r,\\mathcal{B}^r)$\u4e0a\u306e\u6709\u9650\u6e2c\u5ea6$\\mu$\u306b\u5bfe\u3057\u3066, $\\Sigma_l^{r,s}(\\Psi)$\u306f$\\mathbb{M}(\\mathbb{R}^r,\\mathbb{R}^s)$\u306b\u304a\u3044\u3066$\\rho_{\\mu}^s$-\u7a20\u5bc6\u3067\u3042\u308b.\n  \u3055\u3089\u306b, $\\mu(K) = \\mu(\\mathbb{R}^r)$\u306a\u308b\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}^r$\u304c\u5b58\u5728\u3059\u308c\u3070, \u4efb\u610f\u306e$p \\geq 1$\u306b\u5bfe\u3057\u3066, $\\Sigma_l^{r,s}(\\Psi)$\u306f$L^p(r,s,\\mu)$\n  \u306b\u304a\u3044\u3066$d_{\\mu}^{p,s}$-\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{thm}\n\n\\subsection{\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306b\u3088\u308b\u88dc\u9593}\n\u672c\u7bc0\u3067\u306ffeedforward\u578b\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306b\u3088\u308b\u88dc\u9593\u306b\u3064\u3044\u3066\u8ff0\u3079\u308b. \n\u3064\u307e\u308a, \u4e0e\u3048\u3089\u308c\u305f\u6709\u9650\u500b\u306e\u70b9\u3092(\u8fd1\u4f3c\u7684\u307e\u305f\u306f\u771f\u306b)\u901a\u308b\u3088\u3046\u306afeedforward\u578b\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u5b58\u5728\u306b\u3064\u3044\u3066\u8ff0\u3079\u308b. \n\\begin{thm}\n  \u95a2\u6570$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f$\\Sigma^r(\\Psi)$\u304c$C(\\mathbb{R}^r,\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u306b\u306a\u308b\u3082\u306e\u3068\u3059\u308b.  \u307e\u305f$(\\mathbb{R}^r,\\mathcal{B}^r)$\u4e0a\u306e\u6709\u9650\u6e2c\u5ea6$\\mu$\u306f\u6709\u9650\u500b\u306e\u70b9\u306e\u4e0a\u3067\u306e\u307f\u5024\u3092\u53d6\u308b\u3082\u306e\u3068\u3059\u308b. \u3064\u307e\u308a, \u76f8\u7570\u306a\u308b$x_1,\\ldots,x_n \\in \\mathbb{R}^r$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\sum_{j=1}^n \\mu(\\{x_j\\}) = \\mu(\\mathbb{R}^r),~~ \\mu(\\{x_j\\}) > 0 ~~(j=1,\\ldots,n)\n  \\]\n  \u3068\u306a\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$g \\in \\mathbb{M}^r$\u3068$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, $f \\in \\Sigma^r(\\Psi)$\u304c\u5b58\u5728\u3057\u3066, $\\mu(|f-g|>\\varepsilon) = 0$\u3068\u306a\u308b. \n\\end{thm}\n\\begin{proof}\n  $\\tilde{\\varepsilon} := \\min_{1\\leq j \\leq n} \\mu(\\{x_j\\})$\u3068\u304a\u304f. $\\varepsilon < \\tilde{\\varepsilon}$\u3068\u3057\u3066\u3088\u3044. \u5b9a\u7406\\ref{UATinMr}\u3088\u308a, $\\Sigma^r(\\Psi)$\u306f$\\mathbb{M}^r$\u306b\u304a\u3044\u3066$\\rho_{\\mu}$-\u7a20\u5bc6\u3067\u3042\u308b\u306e\u3067, $f \\in \\Sigma^r(\\Psi)$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\rho_{\\mu}(f,g) = \\inf\\{\\varepsilon' > 0 \\mid \\mu(|f-g|> \\varepsilon') < \\varepsilon'\\} < \\varepsilon\n  \\]\n  \u3068\u306a\u308b. \u3053\u306e\u3068\u304d, \u3042\u308b$\\varepsilon' > 0$\u3067, $\\mu(|f-g|> \\varepsilon') < \\varepsilon'$\u304b\u3064$\\varepsilon' < \\varepsilon$\u306a\u308b\u3082\u306e\u304c\u53d6\u308c\u308b. \u3059\u308b\u3068, \n  \\[\n  \\mu(|f-g|> \\varepsilon) \\leq \\mu(|f-g|> \\varepsilon') < \\varepsilon' < \\varepsilon < \\tilde{\\varepsilon}\n  \\]\n  \u3068\u306a\u308b\u306e\u3067, $\\mu(|f-g|> \\varepsilon) = 0$\u3067\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044. \n\\end{proof}\n\\begin{cor}\n  \u95a2\u6570$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f$\\Sigma^r(\\Psi)$\u304c$C(\\mathbb{R}^r,\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u306b\u306a\u308b\u3082\u306e\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e\u95a2\u6570$g:\\{0,1\\}^r \\rightarrow \\{0,1\\}$\u3068$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, $f \\in \\Sigma^r(\\Psi)$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\sup_{x \\in \\{0,1\\}^r} |f(x) - g(x)| < \\varepsilon\n  \\]\n  \u3068\u306a\u308b. \n\\end{cor}\n\\begin{proof}\n  $\\mu$\u3092$\\{0,1\\}^r$\u306e\u5404\u70b9\u3067$1\/2^r$\u3092\u5024\u306b\u53d6\u308b$(\\mathbb{R}^r,\\mathcal{B}^r)$\u4e0a\u306e\u6e2c\u5ea6\u3068\u3059\u308b. $g$(\u6b63\u78ba\u306b\u306f$g$\u3092$\\mathbb{R}^r$\u306b\u62e1\u5f35\u3057\u305f\u3082\u306e)\u306f$\\mathbb{M}^r$\u306b\u5c5e\u3059\u308b\u306e\u3067, \u3059\u3050\u4e0a\u306e\u5b9a\u7406\u3088\u308a$f \\in \\Sigma^r(\\Psi)$\u304c\u5b58\u5728\u3057\u3066, $\\mu(|f-g| \\geq \\varepsilon) = 0$\u3068\u306a\u308b. \u3053\u306e\u3068\u304d, \u3082\u3057$x \\in \\{0,1\\}^r$\u304c\u5b58\u5728\u3057\u3066, $|f(x)-g(x)| \\geq \\varepsilon$\u3068\u306a\u308b\u306a\u3089, $\\mu((|f-g| \\geq \\varepsilon) \\geq \\mu(\\{x\\}) = 1\/2^r > 0$\u3068\u306a\u308a\u4e0d\u5408\u7406\u3067\u3042\u308b. \u3088\u3063\u3066, $\\sup_{x \\in \\{0,1\\}^r} |f(x) - g(x)| < \\varepsilon$\u3067\u3042\u308b. \n\\end{proof}\n\u3055\u3089\u306b, $\\Psi$\u304c\u3042\u308b\u6761\u4ef6\u3092\u6e80\u305f\u305b\u3070, \u4efb\u610f\u306e\u95a2\u6570$g:\\mathbb{R}^r \\rightarrow \\mathbb{R}$\u3068\u4efb\u610f\u306b\u4e0e\u3048\u3089\u308c\u305f\u6709\u9650\u500b\u306e\u70b9\u306b\u5bfe\u3057\u3066, \u305d\u306e\u70b9\u306e\u4e0a\u3067$g$\u3068\u5024\u304c\u4e00\u81f4\u3059\u308b\u3088\u3046\u306a$f \\in \\Sigma^r(\\Psi)$\u304c\u5b58\u5728\u3059\u308b. \u3053\u306e\u3053\u3068\u3092\u793a\u3059\u305f\u3081\u306b\u6b21\u306e\u88dc\u984c\u3092\u793a\u3059. \n\\begin{lem}\n  $n \\in \\mathbb{N}$\u3068\u3057, $K$\u3092$|K| \\geq n+1$\u306a\u308b\u4f53\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$K$\u4e0a\u306e\u7dda\u5f62\u7a7a\u9593$V$\u306f, $k \\leq n$\u500b\u306e\u771f\u7dda\u5f62\u90e8\u5206\u7a7a\u9593\u306e\u548c\u96c6\u5408\u3067\u8868\u305b\u306a\u3044. \u3064\u307e\u308a, \u3069\u3093\u306a\u7dda\u5f62\u90e8\u5206\u7a7a\u9593$V_1 ,\\ldots,V_k \\subsetneq V$\u306b\u3064\u3044\u3066\u3082$V = \\bigcup_{j=1}^k V_j$\u3068\u306f\u306a\u3089\u306a\u3044. \n  \u7279\u306b, $\\mathbb{R}$\u4e0a\u306e\u7dda\u5f62\u7a7a\u9593$V$\u306f\u771f\u7dda\u5f62\u90e8\u5206\u7a7a\u9593\u306e\u6709\u9650\u548c\u96c6\u5408\u3067\u8868\u305b\u306a\u3044. \n\\end{lem}\n\\begin{proof}\n  $n$\u306b\u95a2\u3059\u308b\u5e30\u7d0d\u6cd5\u306b\u3088\u308b. $n = 1$\u306e\u3068\u304d\u306f\u660e\u3089\u304b\u306b\u6210\u7acb\u3059\u308b(\u7dda\u5f62\u7a7a\u9593\u3068\u771f\u7dda\u5f62\u90e8\u5206\u7a7a\u9593\u306f\u4e00\u81f4\u3057\u306a\u3044). $n$\u3067\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b. \n  \u3053\u306e\u3068\u304d, \u3082\u3057\u3042\u308b$|K| \\geq n+2$\u306a\u308b\u4f53$K$\u3068$K$\u4e0a\u306e\u7dda\u5f62\u7a7a\u9593$V$\u304a\u3088\u3073$k \\leq n+1$\u306b\u3064\u3044\u3066, \u7dda\u5f62\u90e8\u5206\u7a7a\u9593$V_1 ,\\ldots,V_k \\subsetneq V$\u304c\u5b58\u5728\u3057\u3066$V = \\bigcup_{j=1}^k V_j$\u3068\u306a\u3063\u305f\u3068\u3059\u308b. \n  \u3053\u3053\u3067\u3059\u3079\u3066\u306e$V_j$\u306f$\\{0\\}$\u3067\u306a\u3044\u3068\u3057\u3066\u3088\u3044. \n  \u3044\u307e, $x \\in V_k \\setminus \\{0\\}$\u3092\u3068\u308a, $y \\in V \\setminus V_k$\u3092\u3068\u308b\u3068, \u4efb\u610f\u306e$\\alpha \\in K \\setminus \\{0\\}$\u306b\u5bfe\u3057\u3066$x + \\alpha y \\in V \\setminus V_k = \\bigcup_{j=1}^{k-1} V_j$\u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, $|K| \\geq n+2 $\u306b\u6ce8\u610f\u3057\u3066\u76f8\u7570\u306a\u308b$\\alpha_1,\\ldots,\\alpha_{k} \\in K \\setminus \\{0\\}$\u3092\u53d6\u308b\u3068, \u9ce9\u306e\u5de3\u539f\u7406\u306b\u3088\u308a, \u3042\u308b$l \\neq k$\u3068$i,j$\u306b\u3064\u3044\u3066$x + \\alpha_{i}y , x + \\alpha_{j}y \\in V_l$\u3068\u306a\u308b. \u3059\u308b\u3068, $V_l$\u306f\u7dda\u5f62\u90e8\u5206\u7a7a\u9593\u3067\u3042\u308b\u306e\u3067, \n  \\[\n  (\\alpha_i - \\alpha_j)y = (x + \\alpha_i y) - (x + \\alpha_j y) \\in V_l\n  \\]\n  \u3068\u306a\u308a, $\\alpha_i - \\alpha_j \\neq 0$\u3086\u3048, \u3053\u306e\u9006\u5143\u3092\u304b\u3051\u308b\u3053\u3068\u3067$y \\in V_l$\u304c\u308f\u304b\u308b. \u3088\u3063\u3066, $x = (x + \\alpha_i y)-\\alpha_i y \\in V_l$\u3067\u3042\u308b. \u4ee5\u4e0a\u304b\u3089, $V = \\bigcup_{j=1}^{k-1} V_j$\u3068\u306a\u308b. \u3053\u308c\u306f\u5e30\u7d0d\u6cd5\u306e\u4eee\u5b9a\u306b\u53cd\u3059\u308b. \u3086\u3048\u306b$n+1$\u306e\u3068\u304d\u3082\u6210\u7acb\u3057\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044. \n\\end{proof}\n\\begin{thm}\n  $\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3092squashing\u95a2\u6570\u3068\u3059\u308b. \u3082\u3057$\\Psi(z_0) = 0, \\Psi(z_1) = 1$\u306a\u308b$z_0,z_1 \\in \\mathbb{R}^r$\u304c\u5b58\u5728\u3059\u308b\u306a\u3089\u3070, \u76f8\u7570\u306a\u308b\u4efb\u610f\u306e\u70b9$x_1,\\ldots,x_n \\in \\mathbb{R}^r$\u3068\u4efb\u610f\u306e\u95a2\u6570$g:\\mathbb{R}^r \\rightarrow \\mathbb{R}$\u306b\u5bfe\u3057\u3066, $f \\in \\Sigma^r(\\Psi)$\u304c\u5b58\u5728\u3057\u3066$f(x_j)=g(x_j) ~(j=1,\\ldots,n)$\u3068\u306a\u308b. \n\\end{thm}\n\\begin{proof}\n  \u4e8c\u6bb5\u968e\u306b\u5206\u3051\u3066\u8a3c\u660e\u3059\u308b. Step1\u3067$r=1$\u306e\u5834\u5408\u3092\u8a3c\u660e\u3057, Step2\u3067$r \\geq 2$\u306e\u5834\u5408\u3092\u8a3c\u660e\u3059\u308b. \\\\\n  {\\bf Step1}: \u6dfb\u5b57\u3092\u4ed8\u3051\u66ff\u3048\u308b\u3053\u3068\u3067$x_1 < \\cdots < x_n$\u3068\u3057\u3066\u3088\u3044. \u3044\u307e, $\\Psi$\u306f\u5358\u8abf\u975e\u6e1b\u5c11\u306a\u306e\u3067$M > 0$\u3092\u9069\u5f53\u306b\u53d6\u308c\u3070\u4efb\u610f\u306e$x \\in \\mathbb{R}$\u306b\u5bfe\u3057\u3066, $x \\leq -M$\u306a\u3089\u3070$\\Psi(x)=0$\u3068\u306a\u308a, $x \\geq M$\u306a\u3089\u3070$\\Psi(x)=1$\u3068\u306a\u308b. \u305d\u3053\u3067, $A_1(x) = M ~(x \\in \\mathbb{R})$\u3068\u3057, $\\beta_1 = g(x_1)$\u3068\u304a\u304f. \u307e\u305f, $A_j \\in \\mathbb{A}^1 ~~(j=2,\\ldots,n)$\u3092$A_j(x_{j-1}) = -M$\u304b\u3064$A_j(x_j) = M$\u3068\u306a\u308b\u3082\u306e\u3068\u3057, $\\beta_j = g(x_j) - g(x_{j-1})$\u3068\u3059\u308b. \u3059\u308b\u3068, \n  \\[\n  f = \\sum_{j=1}^{n} \\beta_j \\Psi \\circ A_j ~\\in \\Sigma^1(\\Psi)\n  \\]\n  \u304c\u6c42\u3081\u308b\u3082\u306e\u3067\u3042\u308b. \\\\\n  {\\bf Step2}: $r \\geq 2$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u3042\u308b$p \\in \\mathbb{R}^r$\u304c\u5b58\u5728\u3057\u3066, $i \\neq j$\u306a\u3089\u3070$p^{\\mathrm{T}}(x_i-x_j) \\neq 0$\u3068\u306a\u308b. \u5b9f\u969b, $\\bigcup_{i \\neq j} \\{ q \\in \\mathbb{R}^r \\mid q^{\\mathrm{T}}(x_i - x_j) = 0 \\}$\u306f$\\mathbb{R}^r$\u306e\u8d85\u5e73\u9762\u306e\u6709\u9650\u548c\u3067\u3042\u308b\u306e\u3067\u3059\u3050\u4e0a\u306e\u88dc\u984c\u3088\u308a$\\mathbb{R}^r$\u3092\u8986\u3044\u5c3d\u304f\u3059\u3053\u3068\u306f\u3067\u304d\u306a\u3044. \u3053\u3053\u3067,  \u6dfb\u5b57\u3092\u4ed8\u3051\u66ff\u3048\u308b\u3053\u3068\u3067$p^{\\mathrm{T}}x_1 < \\cdots < p^{\\mathrm{T}} x_n$\u3068\u3057\u3066\u3088\u3044. \u3059\u308b\u3068, Step1\u3068\u540c\u7b49\u306b\u3057\u3066, $\\beta_j \\in \\mathbb{R}, ~A_j \\in \\mathbb{A}^1$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\sum_{j=1}^n \\beta_j \\Psi(A_j(p^{\\mathrm{T}}x_k)) = g(x_k) ~~~(k=1,\\ldots,n)\n  \\]\n  \u3068\u306a\u308b. $A_j'(x) = A_j(p^{\\mathrm{T}}x) ~(x \\in \\mathbb{R}^r)$\u3068\u304a\u304f\u3068$A_j' \\in \\mathbb{A}^r$\u3067\u3042\u308b\u306e\u3067, \n  \\[\n  f = \\sum_{j=1}^n \\beta_j \\Psi \\circ A_j'\n  \\]\n  \u304c\u6c42\u3081\u308b\u3082\u306e\u3067\u3042\u308b. \n\\end{proof}\n\u6b21\u306e\u5b9a\u7406\u306fPinkus\\cite{Pinkus}\u306b\u3088\u308b. \n\\begin{thm}\n  $\\Psi \\in C(\\mathbb{R})$\u306f\u3042\u308b\u958b\u533a\u9593\u4e0a\u3067\u591a\u9805\u5f0f\u3067\u306a\u3044\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$\\alpha_1,\\ldots,\\alpha_k \\in \\mathbb{R}$\u3068\u76f8\u7570\u306a\u308b\u4efb\u610f\u306e\u70b9$x_1,\\ldots,x_n \\in \\mathbb{R}^r$\u306b\u5bfe\u3057\u3066, $w_1,\\ldots,w_k \\in \\mathbb{R}^r$\u3068$c_1,\\ldots,c_k$,$b_1,\\ldots,b_k \\in \\mathbb{R}$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\sum_{j=1}^k c_j \\Psi(w_j^{\\mathrm{T}}x_i + b_j) = \\alpha_i ~~~~(i=1,\\ldots,k)\n  \\]\n  \u3068\u306a\u308b. \n\\end{thm}\n\\begin{proof}\n  $K := \\{ x_1,\\ldots,x_k\\} \\subset \\mathbb{R}^r$\u3068\u304a\u304f. \u3044\u307e, \u3059\u3079\u3066\u304c$0$\u3068\u306f\u306a\u3089\u306a\u3044$d_1,\\ldots,d_k \\in \\mathbb{R}$\u304c\u5b58\u5728\u3057\u3066\n  \\[\n  \\sum_{i=1}^k d_i \\Psi(w^{\\mathrm{T}} x_i + b) = 0 ~~~~(\\forall w \\in \\mathbb{R}^r, b \\in \\mathbb{R})\n  \\]\n  \u3068\u306a\u308b\u3068\u4eee\u5b9a\u3057\u3088\u3046. \u3053\u306e\u3068\u304d, \n  \\[\n  G: C(K) \\ni f \\mapsto \\sum_{i=1}^k d_i f(x_i) \\in \\mathbb{R}\n  \\]\n  \u3068\u304a\u304f\u3068, \u3042\u308b$d_i$\u306f$0$\u3067\u306a\u3044\u306e\u3067$G$\u306f$0$\u3067\u306a\u3044\u6709\u754c\u7dda\u5f62\u6c4e\u95a2\u6570\u3067\u3042\u308a, \n  \\[\n  G(f) = 0 ~~~~(\\forall f \\in \\mathrm{span}\\{K \\ni x \\mapsto \\Psi(w^{\\mathrm{T}}x+b) \\mid w \\in \\mathbb{R}^r, b \\in \\mathbb{R}\\})\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, $\\mathrm{span}\\{K \\ni x \\mapsto \\Psi(w^{\\mathrm{T}}x+b) \\mid w \\in \\mathbb{R}^r, b \\in \\mathbb{R}\\}$\u306f$C(K)$\u306b\u304a\u3044\u3066\u7a20\u5bc6\u3067\u306a\u3044(\u7a20\u5bc6\u306a\u3089$G=0$\u3067\u3051\u308c\u3070\u306a\u3089\u306a\u3044). \u3053\u308c\u306f\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406(\u5b9a\u7406\\ref{LeshnoMainResult1})\u306b\u53cd\u3059\u308b. \n  \u3088\u3063\u3066, $w,b$\u306e\u9023\u7d9a\u95a2\u6570$\\Psi(w^{\\mathrm{T}} x_1 + b), \\ldots, \\Psi(w^{\\mathrm{T}} x_k + b)$\u306f\u4e00\u6b21\u72ec\u7acb\u3067\u3042\u308b. \u3086\u3048\u306b, $w_1,\\ldots,w_k \\in \\mathbb{R}^r$\u53ca\u3073$b_1,\\ldots,b_k$\u304c\u5b58\u5728\u3057\u3066\n  \\[\n  \\mathrm{det}((\\Psi(w_j x_i + b_j))_{i,j=1,\\ldots,k}) \\neq 0\n  \\]\n  \u3068\u306a\u308b. \u3086\u3048\u306b, $c_1,\\ldots,c_k \\in \\mathbb{R}$\u306b\u95a2\u3059\u308b\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\n  \\[\n  \\sum_{j=1}^k c_j \\Psi(w_j^{\\mathrm{T}}x_i + b_j) = \\alpha_i ~~~~(i=1,\\ldots,k)\n  \\]\n  \u306f\u89e3\u3092\u6301\u3064. \n\\end{proof}\n\n\\numberwithin{thm}{section}\n\\section{RBF\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406}\n\u672c\u7ae0\u3067\u306f\u4ee5\u4e0b\u3067\u5b9a\u7fa9\u3059\u308bRBF(Radial-Basis-Function)\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u4e07\u80fd\u8fd1\u4f3c\u80fd\u529b\u306b\u3064\u3044\u3066\u8ff0\u3079\u308b. \u3059\u306a\u308f\u3061, \u95a2\u6570$K$\u304c\u3042\u308b\u6761\u4ef6\u3092\u307f\u305f\u3059\u3068\u304d\u306b$K$\u3092\u57fa\u5e95\u95a2\u6570\u3068\u3059\u308bRBF\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u304c$L^p(\\mathbb{R}^r)$\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3084, $C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059. \n\n\\begin{defn}\n  \u95a2\u6570$K:\\mathbb{R}^r \\rightarrow \\mathbb{R}$\u306b\u5bfe\u3057\u3066, \n  \\[\n  q(x) = \\sum_{i=1}^M w_i K\\left(\\frac{x-z_i}{\\sigma}\\right) ~~~(M \\in \\mathbb{N}, w_i \\in \\mathbb{R}, \\sigma > 0 , z_i \\in \\mathbb{R}^r)\n  \\]\n  \u3068\u3044\u3046\u5f62\u306e\u95a2\u6570$q:\\mathbb{R}^r \\rightarrow \\mathbb{R}$\u5168\u4f53\u306e\u96c6\u5408\u3092$S_K$\u3068\u8868\u3057, $S_K$\u3092$K$\u3092\u57fa\u5e95\u95a2\u6570\u3068\u3059\u308bRBF\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3068\u3044\u3046. \n\\end{defn}\n \u307e\u305a\u6b21\u306e\u88dc\u984c\u3092\u7528\u610f\u3057\u3066\u304a\u304f. \n\\begin{lem}\n  $f \\in L^p(\\mathbb{R}^r), ~p \\in [1,\\infty)$\u3068\u3057, $\\phi \\in L^1(\\mathbb{R}^r)$\u306f$\\int_{\\mathbb{R}^r} \\phi(x) dx = 1$\u3092\u307f\u305f\u3059\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\phi_{\\varepsilon}(x) := \\frac{1}{\\varepsilon^r}\\phi\\left(\\frac{x}{\\varepsilon}\\right) ~~(x \\in \\mathbb{R}^r)\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b\u3068, $\\lVert \\phi_{\\varepsilon} * f - f \\rVert_{L^p} \\rightarrow 0 ~~(\\varepsilon \\rightarrow 0)$\u3068\u306a\u308b. \n\\end{lem}\n\\begin{proof}\n  $\\phi_{\\varepsilon} \\in L^1(\\mathbb{R}^r) ~~(\\varepsilon > 0)$\u3067\u3042\u308b\u306e\u3067, \u4ed8\u9332\u306e\u547d\u984c\\ref{L1andLpConvolution}\u3088\u308a$\\phi_{\\varepsilon}*f \\in L^p(\\mathbb{R}^r)$\u3067\u3042\u308b. \u307e\u305f, $\\mathrm{a.e.}y \\in \\mathbb{R}^r$\u306b\u5bfe\u3057\u3066, \u5909\u6570\u5909\u63db\u306b\u3088\u308a, \n  \\[\n  (\\phi_{\\varepsilon}*f)(y) = \\int_{\\mathbb{R}^r} \\phi_{\\varepsilon}(x) f(y-x) dx = \\int_{\\mathbb{R}^r} f(y-\\varepsilon x) \\phi(x) dx\n  \\]\n  \u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, $\\int_{\\mathbb{R}^r} \\phi(x) dx = 1$\u3088\u308a, \n  \\[\n  (\\phi_{\\varepsilon} * f)(y) - f(y) = \\int_{\\mathbb{R}^r} ( f(y-\\varepsilon x) - f(y) ) \\phi(x) dx\n  \\]\n  \u3067\u3042\u308b. \u305d\u3053\u3067, $q \\in [1,\\infty]$\u3092$1\/p + 1\/q = 1$\u3092\u307f\u305f\u3059\u3082\u306e\u3068\u3059\u308b\u3068, H\\\"{o}lder\u306e\u4e0d\u7b49\u5f0f(\u4ed8\u9332\u306e\u547d\u984c\\ref{HolderInequality})\u3088\u308a, \n  \\[\n  \\begin{aligned}\n  \\left\\lvert (\\phi_{\\varepsilon} * f)(y) - f(y) \\right\\rvert\n  &\\leq \\int_{\\mathbb{R}^r} \\left\\lvert f(y-\\varepsilon x) - f(y) \\right\\rvert \\lvert \\phi(x) \\rvert dx  \\\\\n  &= \\int_{\\mathbb{R}^r} \\left\\lvert f(y-\\varepsilon x) - f(y) \\right\\rvert \\lvert \\phi(x) \\rvert^{1-1\/q} \\lvert \\phi(x) \\rvert^{1\/q} dx \\\\\n  &\\leq \\lVert |\\phi|^{1\/q} \\rVert_{L^q} \\left(\\int_{\\mathbb{R}^r} |f(y-\\varepsilon x) - f(y)|^p |\\phi(x)| dx \\right)^{1\/p}\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b($p=1$\u306e\u3068\u304d\u306f$\\lVert |\\phi|^{1\/q} \\rVert_{L^q} = 1$\u3068\u89e3\u91c8\u3059\u308b). \u3086\u3048\u306b, Fubini\u306e\u5b9a\u7406\u3088\u308a, \n  \\[\n  \\begin{aligned}\n  \\lVert \\phi_{\\varepsilon}*f - f \\rVert_{L^p}^p \n  &\\leq \\lVert |\\phi|^{1\/q} \\rVert_{L^q}^p \\int_{\\mathbb{R}^r} \\int_{\\mathbb{R}^r} \\lvert f(y-\\varepsilon x) - f(y) \\rvert^p \\lvert \\phi(x) \\rvert dx dy \\\\\n  &= \\lVert |\\phi|^{1\/q} \\rVert_{L^q}^p \\int_{\\mathbb{R}^r} \\lvert \\phi(x) \\rvert \\lVert f(\\cdot - \\varepsilon x) - f(\\cdot) \\rVert_{L^p}^p dx\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u4efb\u610f\u306e$x \\in \\mathbb{R}^r, \\varepsilon > 0$\u306b\u5bfe\u3057\u3066, \u4e09\u89d2\u4e0d\u7b49\u5f0f\u3088\u308a\n  \\[\n  \\begin{aligned}\n  \\lvert \\phi(x) \\rvert \\lVert f(\\cdot - \\varepsilon x) - f(\\cdot) \\rVert_{L^p} \\leq 2 \\lvert  \\phi(x) \\rvert  \\lVert f \\rVert_{L^p}\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308a, $L^p$\u30ce\u30eb\u30e0\u306e\u5e73\u884c\u79fb\u52d5\u306b\u95a2\u3059\u308b\u9023\u7d9a\u6027(\u4ed8\u9332\u306e\u547d\u984c\\ref{ContinuousOfTranslationAboutLpNorm})\u3088\u308a, \n  \\[\n  \\lVert f(\\cdot - \\varepsilon x) - f(\\cdot) \\rVert_{L^p} \\rightarrow 0 ~~(\\varepsilon \\rightarrow 0)\n  \\]\n  \u3067\u3042\u308b\u304b\u3089, \u512a\u53ce\u675f\u5b9a\u7406\u3088\u308a\n  \\[\n  \\lVert \\phi_{\\varepsilon}*f - f \\rVert_{L^p} \\rightarrow 0 ~~(\\varepsilon \\rightarrow 0)\n  \\]\n  \u3067\u3042\u308b. \n\\end{proof}\n\\begin{thm}[J. Parkn and I. W. Sandberg, 1991 \\cite{Park}] \\\\\n  $\\mu$\u3092$\\mathbb{R}^r$\u4e0a\u306eLebesgue\u6e2c\u5ea6\u3068\u3059\u308b. \u95a2\u6570$K:\\mathbb{R}^r \\rightarrow \\mathbb{R}$\u304c\u4ee5\u4e0b\u3092\u6e80\u305f\u3059\u3068\u3059\u308b. \n  \\begin{itemize}\n      \\item[(1)] $K$\u306f\u6709\u754c. \n      \\item[(2)] $K \\in L^1(\\mathbb{R}^r,\\mu)$\u304b\u3064$\\int_{\\mathbb{R}^r} K d\\mu \\neq 0$.\n      \\item[(3)] $\\mu(\\{K\\mbox{\u306e\u4e0d\u9023\u7d9a\u70b9}\\}) = 0$\n  \\end{itemize}\n  \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$p \\geq 1$\u306b\u5bfe\u3057\u3066, $S_K$\u306f$L^p(\\mathbb{R}^r,\\mu)$\u306b\u304a\u3044\u3066$L^p$\u30ce\u30eb\u30e0\u306b\u3064\u3044\u3066\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{thm}\n\\begin{proof}\n  \u4efb\u610f\u306b$f \\in L^p(\\mathbb{R}^r,\\mu)$\u3068$\\varepsilon > 0$\u3092\u3068\u308b. $C_0(\\mathbb{R}^r)$\u306f$L^p(\\mathbb{R}^r,\\,u)$\u306b\u304a\u3044\u3066$L^p$\u30ce\u30eb\u30e0\u306b\u3064\u3044\u3066\u7a20\u5bc6\u3067\u3042\u308b\u304b\u3089, $f_c \\in C_0(\\mathbb{R}^r)$\u304c\u5b58\u5728\u3057\u3066, $\\lVert f - f_c \\rVert_{L^p} < \\varepsilon\/3$\u3068\u306a\u308b. $0 \\in S_K$\u3067\u3042\u308b\u304b\u3089$f_c \\neq 0$\u3068\u4eee\u5b9a\u3057\u3066\u3088\u3044. \u3053\u3053\u3067\u95a2\u6570$\\phi:\\mathbb{R}^r \\rightarrow \\mathbb{R}$\u3092\n  \\[\n  \\phi(x) = \\frac{1}{\\int_{\\mathbb{R}^r} K d\\mu} K(x) \n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u305d\u3057\u3066, $\\sigma > 0$\u306b\u5bfe\u3057\u3066, $\\phi_{\\sigma}(x) = \\phi(x\/\\sigma)\/\\sigma^r$\u3068\u5b9a\u7fa9\u3059\u308b. \u3053\u306e\u3068\u304d, $\\lVert \\phi_{\\sigma}*f_c - f_c \\rVert_{L^p} \\rightarrow 0 ~(\\sigma \\rightarrow 0)$\u3068\u306a\u308b. \u305d\u3053\u3067, $\\sigma > 0$\u3092$\\lVert \\phi_{\\sigma}*f_c - f_c \\rVert_{L^p} < \\varepsilon\/3$\u3068\u306a\u308b\u3088\u3046\u306b\u3068\u3063\u3066\u304a\u304f. \u6b21\u306b$\\mathrm{supp}f_c \\subset [-T,T]^r$\u306a\u308b$T > 0$\u3092\u3068\u308b. \u5404$\\alpha \\in \\mathbb{R}^r$\u306b\u5bfe\u3057\u3066$\\phi_{\\sigma}(\\alpha - \\cdot) f_c(\\cdot)$\u306f\u6709\u754c\u304b\u3064\u307b\u3068\u3093\u3069\u81f3\u308b\u6240\u9023\u7d9a\u3067\u3042\u308b\u304b\u3089$[-T,T]^r$\u4e0aRiemann\u7a4d\u5206\u53ef\u80fd\u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, \u5404$[-T,T]$\u3092$n$\u7b49\u5206\u3059\u308b\u3053\u3068\u3067\u5f97\u3089\u308c\u308b$[-T,T]^r$\u306e\u5206\u5272\u3092$\\Delta_n$\u3068\u304a\u304d, $n^r$\u500b\u3042\u308b$\\Delta_n$\u306e\u5404\u5c0f\u9818\u57df\u304b\u3089\u4efb\u610f\u306b$\\alpha_i$\u3092\u9078\u3076\u3068, \n  \\[\n  v_n(\\alpha) := \\sum_{i=1}^{n^r} \\phi_{\\sigma}(\\alpha - \\alpha_i)f_c(\\alpha_i)\\left(\\frac{2T}{n}\\right)^r\n  \\]\n  \u306f$\\int_{[-T,T]^r} \\phi_{\\sigma}(\\alpha - x)f_c(x)dx$\u306b\u53ce\u675f\u3059\u308b. \u4e00\u65b9, $\\mathrm{supp}f_c \\subset [-T,T]^r$\u3067\u3042\u308b\u306e\u3067, \n  \\[\n  \\int_{[-T,T]^r} \\phi_{\\sigma}(\\alpha - x)f_c(x)dx = \\int_{\\mathbb{R}^r} \\phi_{\\sigma}(\\alpha - x)f_c(x)dx = (\\phi_{\\sigma} * f_c)(\\alpha)\n  \\]\n  \u3067\u3042\u308b. \u3088\u3063\u3066, \u5404\u70b9$\\alpha \\in \\mathbb{R}^r$\u306b\u3064\u3044\u3066$v_n(\\alpha) \\rightarrow (\\phi_{\\sigma} * f_c)(\\alpha)$\u3068\u306a\u308b. $\\phi_{\\sigma}*f_c \\in L^p(\\mathbb{R}^r,\\mu)$\u306b\u6ce8\u610f\u3059\u308b\u3068, \u3042\u308b$T_1 > 0$\u306b\u3064\u3044\u3066, \n  \\[\n  \\int_{\\mathbb{R}^r \\setminus [-T_1,T_1]^r} |\\phi_{\\sigma}*f_c|^p d\\mu < (\\varepsilon\/9)^p\n  \\]\n  \u3068\u306a\u308b. \u307e\u305f, $\\phi_{\\sigma} \\in L^1(\\mathbb{R}^r,\\mu)$\u3067\u3042\u308a, \u304b\u3064$\\phi_{\\sigma}$\u306f\u6709\u754c\u3067\u3042\u308b\u3053\u3068\u3088\u308a$|\\phi_{\\sigma}|^p \\in L^1$\u3068\u306a\u308b\u304b\u3089, \u3042\u308b$T_2 > 0$\u306b\u3064\u3044\u3066, \n  \\[\n  \\int_{\\mathbb{R}^r \\setminus [-T_2,T_2]^r} |\\phi_{\\sigma}|^p d\\mu < \\left( \\frac{\\varepsilon}{9\\lVert f_c \\rVert_{L^{\\infty}} (2T)^r} \\right)^p\n  \\]\n  \u3068\u306a\u308b. $[0,\\infty) \\ni y \\rightarrow y^p \\in \\mathbb{R}$\u304c\u51f8\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u3088\u308a, \u4efb\u610f\u306e$\\alpha \\in \\mathbb{R}^r$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\left(\\frac{1}{n^r} \\sum_{i=1}^{n^r} |\\phi_{\\sigma}(\\alpha - \\alpha_i)| \\right)^p \\leq \\frac{1}{n^r} \\sum_{i=1}^{n^r} |\\phi_{\\sigma}(\\alpha - \\alpha_i)|^p\n  \\]\n  \u3068\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u3068, \n  \\[\n  |v_n(\\alpha)|^p \\leq (\\lVert f_c \\rVert_{L^{\\infty}} (2T)^r)^p \\frac{1}{n^r} \\sum_{i=1}^{n^r} |\\phi_{\\sigma}(\\alpha - \\alpha_i)|^p\n  \\]\n  \u3067\u3042\u308b. \u3053\u3053\u3067, $T_0 = \\max\\{T_1,T_2+T\\}$\u3068\u304a\u304f\u3068, $\\alpha_i$\u306e\u5404\u6210\u5206$\\alpha_{i,j} $\u306f$|\\alpha_{i,j}| \\leq T$\u3092\u307f\u305f\u3059\u306e\u3067, \n  \\[\n  \\int_{[-T_2,T_2]^r} |\\phi_{\\sigma}(\\alpha)|^p d\\alpha \\leq \\int_{[-T_0,T_0]^r} |\\phi_{\\sigma}(\\alpha - \\alpha_i)|^p d\\alpha\n  \\]\n  \u3068\u306a\u308b. \u3057\u305f\u304c\u3063\u3066, \n  \\[\n  \\int_{\\mathbb{R}^r \\setminus [-T_0,T_0]^r} |\\phi_{\\sigma}(\\alpha - \\alpha_i)|^p d\\alpha \n  \\leq \\int_{\\mathbb{R}^r \\setminus [-T_2,T_2]^r} |\\phi_{\\sigma}(\\alpha)|^p d\\alpha\n  \\]\n  \u3067\u3042\u308b\u306e\u3067, \n  \\[\n  \\begin{aligned}\n  &\\int_{\\mathbb{R}^r \\setminus [-T_0,T_0]^r} |v_n(\\alpha)|^p d\\alpha \\\\\n  &\\leq (\\lVert f_c \\rVert_{L^{\\infty}} (2T)^r)^p \\frac{1}{n^r} \\sum_{i=1}^{n^r} \\int_{\\mathbb{R}^r \\setminus [-T_0,T_0]^r}|\\phi_{\\sigma}(\\alpha - \\alpha_i)|^p d\\alpha \\\\\n  &\\leq (\\lVert f_c \\rVert_{L^{\\infty}} (2T)^r)^p \\frac{1}{n^r} \\sum_{i=1}^{n^r} \\int_{\\mathbb{R}^r \\setminus [-T_2,T_2]^r} |\\phi_{\\sigma}(\\alpha)|^p d\\alpha \\\\\n  &< (\\varepsilon\/9)^p\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u307e\u305f, $T_0 \\geq T_1$\u3088\u308a, \n  \\[\n  \\int_{\\mathbb{R}^r \\setminus [-T_0,T_0]^r} |\\phi_{\\sigma}*f_c|^p d\\mu < (\\varepsilon\/9)^p\n  \\]\n  \u3067\u3042\u308b. \u3044\u307e, $\\phi_{\\sigma}*f_c \\in L^p$\u3068$v_n$\u306e\u6709\u754c\u6027\u3088\u308a\u512a\u53ce\u675f\u5b9a\u7406\u304c\u4f7f\u3048\u3066, \n  \\[\n  \\int_{[-T_0,T_0]^r} |\\phi_{\\sigma}*f_c - v_n|^p d\\mu \\rightarrow 0 ~~~(n \\rightarrow \\infty)\n  \\]\n  \u3068\u306a\u308b. \u3086\u3048\u306b, \u9069\u5f53\u306b$N \\in \\mathbb{N}$\u3092\u53d6\u308b\u3068, \n  \\[\n  \\int_{[-T_0,T_0]^r} |\\phi_{\\sigma}*f_c - v_N|^p d\\mu < (\\varepsilon\/9)^p\n  \\]\n  \u3068\u306a\u308b. \u4ee5\u4e0a\u304b\u3089, \n  \\[\n  \\begin{aligned}\n  \\lVert v_N - \\phi_{\\sigma}*f_c \\rVert_{L^p}\n  &\\leq \\lVert v_N\\cdot \\chi_{\\mathbb{R}^r \\setminus [-T_0,T_0]^r} \\rVert_{L^p} + \\lVert (v_N - \\phi_{\\sigma}*f_c)\\cdot \\chi_{[-T_0,T_0]^r} \\rVert_{L^p} \\\\\n  &~~~ + \\lVert (\\phi_{\\sigma}*f_c) \\cdot \\chi_{\\mathbb{R}^r \\setminus [-T_0,T_0]^r} \\rVert_{L^p} \\\\\n  &< \\varepsilon\/9 + \\varepsilon\/9 + \\varepsilon\/9 = \\varepsilon\/3\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, \n  \\[\n  \\lVert f - v_N \\rVert_{L^p} \\leq \\lVert f - f_c \\rVert_{L^p} + \\lVert f_c - \\phi_{\\sigma}*f_c \\rVert_{L^p} + \\lVert \\phi_{\\sigma}*f_c - v_N \\rVert_{L^p} < \\varepsilon\n  \\]\n  \u3067\u3042\u308b. \u305d\u3057\u3066, \n  \\[\n  v_N = \\sum_{i=1}^{N^r} \\phi_{\\sigma}(\\cdot - \\alpha_i)f_c(\\alpha_i)\\left(\\frac{2T}{N}\\right)^r = \\sum_{i=1}^{N^r} w_i K\\left(\\frac{\\cdot - \\alpha_i}{\\sigma}\\right) \\in S_K\n  \\]\n  \u3067\u3042\u308b. \u305f\u3060\u3057, \n  \\[\n  w_i = \\frac{1}{\\sigma^r}f_c(\\alpha_i)\\left(\\frac{2T}{N}\\right)^r \\frac{1}{\\int_{\\mathbb{R}^r} K d\\mu}\n  \\]\n  \u3067\u3042\u308b. \u3053\u308c\u3067\u793a\u305b\u305f. \n\\end{proof}\n\\begin{thm}\n  $\\mu$\u3092$\\mathbb{R}^r$\u4e0a\u306eLebesgue\u6e2c\u5ea6\u3068\u3059\u308b. \u95a2\u6570$K:\\mathbb{R}^r \\rightarrow \\mathbb{R}$\u304c\u4ee5\u4e0b\u3092\u6e80\u305f\u3059\u3068\u3059\u308b. \n  \\begin{itemize}\n      \\item[(1)] $K$\u306f\u6709\u754c. \n      \\item[(2)] $K \\in L^1(\\mathbb{R}^r,\\mu)$\u304b\u3064$\\int_{\\mathbb{R}^r} K d\\mu \\neq 0$.\n      \\item[(3)] $K$\u306f\u9023\u7d9a\u95a2\u6570\u3067\u3042\u308b. \n  \\end{itemize}\n  \u3053\u306e\u3068\u304d, $S_K$\u306f$C(\\mathbb{R}^r,\\mathbb{R})$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{thm}\n\\begin{proof}\n  \u307e\u305a, \u5148\u3068\u540c\u69d8\u306b, \u95a2\u6570$\\phi:\\mathbb{R}^r \\rightarrow \\mathbb{R}$\u3092\n  \\[\n  \\phi(x) = \\frac{1}{\\int_{\\mathbb{R}^r} K d\\mu} K(x) \n  \\]\n  \u3068\u5b9a\u7fa9\u3057, $\\sigma > 0$\u306b\u5bfe\u3057\u3066, $\\phi_{\\sigma}(x) = \\phi(x\/\\sigma)\/\\sigma^r$\u3068\u5b9a\u7fa9\u3059\u308b. \u4efb\u610f\u306b$M > 0$\u3092\u3068\u308b. $[-M,M]^r$\u4e0a\u306e\u95a2\u6570$f$\u3068$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, $g \\in S_K$\u304c\u5b58\u5728\u3057\u3066, \\\\\n  $\\sup_{x \\in [-M,M]^r} |f(x) - g(x)| < \\varepsilon$\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u5341\u5206\u3067\u3042\u308b. $T > M$\u3092\u3068\u308b\u3068, Tietze\u306e\u62e1\u5f35\u5b9a\u7406\u3088\u308a\n  \\[\n  \\tilde{f}(x) = f(x) ~(x \\in [-M,M]^r), ~~\\tilde{f}(x) = 0 ~(x \\in \\mathbb{R}^r \\setminus [-T,T])\n  \\]\n  \u306a\u308b\u9023\u7d9a\u95a2\u6570$\\tilde{f}:\\mathbb{R}^r \\rightarrow \\mathbb{R}$\u304c\u3068\u308c\u308b. \u660e\u3089\u304b\u306b$\\tilde{f}$\u306f\u4e00\u69d8\u9023\u7d9a\u3067\u3042\u308b\u306e\u3067, \u3042\u308b$\\delta > 0$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\forall x,y \\in \\mathbb{R}^r , |x-y| < \\delta \\Rightarrow |\\tilde{f}(x) - \\tilde{f}(y)| < \\frac{\\varepsilon}{4\\lVert \\phi \\rVert_{L^1}} \n  \\]\n  \u3068\u306a\u308b. \u4e00\u65b9, $\\phi \\in L^1(\\mathbb{R}^r,\\mu)$\u306a\u306e\u3067, \u3042\u308b$T_0 > 0$\u306b\u3064\u3044\u3066, \n  \\[\n  \\int_{\\mathbb{R}^r \\setminus [-T_0,T_0]^r} |\\phi| d\\mu <  \\frac{\\varepsilon}{8\\lVert \\tilde{f} \\rVert_{L^{\\infty}}}\n  \\]\n  \u3068\u306a\u308b. \u305d\u3053\u3067, \u9069\u5f53\u306b$\\sigma > 0$\u3092\u3068\u308a, $\\forall x \\in [-T_0,T_0]^r,~|\\sigma x | < \\delta$\u3068\u306a\u308b\u3088\u3046\u306b\u3059\u308b\u3068, \u4efb\u610f\u306e$\\alpha \\in [-M,M]^r$\u306b\u3064\u3044\u3066, \n  \\[\n  \\begin{aligned}\n  |(\\phi_{\\sigma} * \\tilde{f})(\\alpha) - \\tilde{f}(\\alpha)|\n  &= \\left| \\int_{\\mathbb{R}^r} \\frac{1}{\\sigma^r}\\phi\\left( \\frac{x}{\\sigma} \\right) \\tilde{f}(\\alpha - x) dx - \\int_{\\mathbb{R}^r} \\tilde{f}(\\alpha)\\phi(x) dx \\right| \\\\\n  &= \\left| \\int_{\\mathbb{R}^r} (\\tilde{f}(\\alpha - \\sigma x) - \\tilde{f}(\\alpha))\\phi(x) dx \\right| \\\\\n  &\\leq \\int_{\\mathbb{R}^r} |\\tilde{f}(\\alpha - \\sigma x) - \\tilde{f}(\\alpha)|\\cdot|\\phi(x)| dx \\\\\n  &\\leq \\int_{[-T_0,T_0]^r} |\\tilde{f}(\\alpha - \\sigma x) - \\tilde{f}(\\alpha)|\\cdot|\\phi(x)| dx \\\\\n  &~~~ + \\int_{\\mathbb{R}^r \\setminus [-T_0,T_0]^r} 2\\lVert \\tilde{f} \\rVert_{L^{\\infty}} |\\phi(x)| dx  \\\\\n  &< \\varepsilon\/2\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3053\u3053\u3067, \u524d\u5b9a\u7406\u306e\u8a3c\u660e\u3068\u540c\u69d8\u306b, $[-T,T]^r$\u3092\u7b49\u5206\u5272\u3057, \u5404$\\alpha \\in \\mathbb{R}^r$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\tilde{v}_n(\\alpha) = \\sum_{i=1}^{n^r} \\phi_{\\sigma}(\\alpha - \\alpha_i)\\tilde{f}(\\alpha_i)\\left(\\frac{2T}{n}\\right)^r\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u95a2\u6570$(s,x) \\mapsto \\phi_{\\sigma}(s-x)\\tilde{f}(x)$\u306f$[-M,M]^r \\times [-T,T]^r$\u4e0a\u3067\u4e00\u69d8\u9023\u7d9a\u306a\u306e\u3067, \u3042\u308b$\\delta_0 > 0$\u304c\u5b58\u5728\u3057\u3066, \u4efb\u610f\u306e$s \\in [-M,M]^r, x,y \\in [-T,T]^r$\u306b\u3064\u3044\u3066, $|x-y| < \\delta_0$\u306a\u3089\u3070, \n  \\[\n  |\\phi_{\\sigma}(s-x)\\tilde{f}(x) - \\phi_{\\sigma}(s-y)\\tilde{f}(y)| < \\frac{\\varepsilon}{2(2T)^r}\n  \\]\n  \u3068\u306a\u308b. $N$\u3092\u5341\u5206\u5927\u304d\u304f\u53d6\u308b\u3068, $\\Delta_N$\u306e\u5404\u5c0f\u9818\u57df\u306e\u4efb\u610f\u306e$2$\u70b9\u9593\u306e\u8ddd\u96e2\u306f$\\delta_0$\u672a\u6e80\u306b\u306a\u308b. \u3086\u3048\u306b, $[-T,T]^r = \\bigcup_{i=1}^{N^r} J_i$\u3068\u8868\u3059\u3053\u3068\u306b\u3059\u308b\u3068, \u4efb\u610f\u306e$\\alpha \\in [-M,M]^r$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  &|v_N(\\alpha) - \\int_{[-T,T]^r} \\phi_{\\sigma}(\\alpha - x) \\tilde{f}(x) dx| \\\\\n  &\\leq \\sum_{i=1}^{N^r} \\int_{J_i} |\\phi_{\\sigma}(\\alpha-\\alpha_i) \\tilde{f}(\\alpha_i) -  \\phi_{\\sigma}(\\alpha-x) \\tilde{f}(x)| dx \\\\\n  &\\leq \\sum_{i=1}^{N^r} \\int_{J_i} \\frac{\\varepsilon}{2(2T)^r} dx \\leq \\varepsilon\/2\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u4efb\u610f\u306e$\\alpha \\in [-M,M]^r$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\int_{[-T,T]^r} \\phi_{\\sigma}(\\alpha - x) \\tilde{f}(x) dx = (\\phi_{\\sigma} * \\tilde{f})(\\alpha)\n  \\]\n  \u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u3068, \n  \\[\n  \\begin{aligned}\n  |f(\\alpha) - v_N(\\alpha)| \n  &= |\\tilde{f}(\\alpha) -  v_N(\\alpha)| \\\\\n  &\\leq |\\tilde{f}(\\alpha) - (\\phi_{\\sigma} * \\tilde{f})(\\alpha)| + |(\\phi_{\\sigma} * \\tilde{f})(\\alpha) - v_N(\\alpha)| \\\\\n  &< \\varepsilon\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, $\\sup_{x \\in [-M,M]^r} |f(x) - v_N(x)| \\leq \\varepsilon$\u3067\u3042\u308b. \u3053\u308c\u3067\u793a\u305b\u305f. \n\\end{proof}\n\u306a\u304a, \u4ee5\u4e0a\u306e\u7d50\u679c\u306b\u95a2\u9023\u3057\u3066, Pinkus\u306b\u3088\u308a\u6b21\u306e\u7d50\u679c\u304c\u5f97\u3089\u308c\u3066\u3044\u308b. \n\\begin{thm}[Pinkus 1996 \\cite{Pinkus1996} Theorem12]\n  $\\sigma \\in C(\\mathbb{R}_{+})$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\mathrm{span}\\{x \\mapsto \\sigma(\\rho \\lVert x - a \\rVert) \\mid \\rho >0, a \\in \\mathbb{R}^r\\}\n  \\]\n  \u304c$C(\\mathbb{R}^r)$\u306b\u304a\u3044\u3066\u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3068, $\\sigma$\u304c\u5076\u591a\u9805\u5f0f\u3068\u4e00\u81f4\u3057\u306a\u3044\u3053\u3068\u306f\u540c\u5024\u3067\u3042\u308b. \n\\end{thm}\n\n\\numberwithin{thm}{subsection}\n\\section{\u8fd1\u4f3c\u30ec\u30fc\u30c8\u306e\u8a55\u4fa1}\n\u672c\u7ae0\u3067\u306f, \u4e0e\u3048\u3089\u308c\u305f\u95a2\u6570\u7a7a\u9593\u306e\u90e8\u5206\u96c6\u5408\u3092$3$\u5c64\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306b\u3088\u308a\u8fd1\u4f3c\u3059\u308b\u3053\u3068\u3092\u8003\u3048\u308b\u3068\u304d, \u4e2d\u9593\u5c64\u306e\u30e6\u30cb\u30c3\u30c8\u6570\u306b\u95a2\u3057\u3066\u3069\u308c\u304f\u3089\u3044\u306e\u52b9\u7387\u3067\u8fd1\u4f3c\u8aa4\u5dee\u304c\u6e1b\u5c11\u3057\u3066\u3044\u304f\u304b\u3068\u3044\u3046\u554f\u984c\u3092\u8003\u3048\u308b. \u3064\u307e\u308a, $X$\u3092\n\\[\n\\mathcal{N}_r^{n}(\\Psi) := \\left\\{x \\mapsto \\sum_{i=1}^n c_i \\Psi(w_i^{\\mathrm{T}} x + b_i) ~\\vline~ w_i \\in \\mathbb{R}^r, c_i,b_i \\in \\mathbb{R} \\right\\}\n\\]\n\u3092\u542b\u3080\u95a2\u6570\u7a7a\u9593\u3068\u3059\u308b\u3068\u304d, \u90e8\u5206\u96c6\u5408$\\mathcal{F} \\subset X$\u306b\u5bfe\u3057\u3066, \u8fd1\u4f3c\u8aa4\u5dee\n\\[\n\\sup_{f \\in \\mathcal{F}} \\inf_{g \\in \\mathcal{N}_r^{n}(\\Psi)} \\lVert f- g \\rVert_X\n\\]\n\u306f$n$\u3092\u5927\u304d\u304f\u3057\u3066\u3044\u304f\u3068\u304d\u3069\u306e\u3088\u3046\u306a\u30ec\u30fc\u30c8\u3067\u6e1b\u5c11\u3059\u308b\u304b?\u3068\u3044\u3046\u554f\u984c\u3092\u8003\u3048\u308b. \n\\subsection{Sobolev\u7a7a\u9593\u306b\u304a\u3051\u308b\u8fd1\u4f3c\u30ec\u30fc\u30c8}\n\u672c\u7bc0\u3067\u306fPinkus,1999 \\cite{Pinkus}\u3067\u7d39\u4ecb\u3055\u308c\u3066\u3044\u308b, Sobolev\u7a7a\u9593\u306b\u304a\u3051\u308b\u8fd1\u4f3c\u30ec\u30fc\u30c8\u306e\u7d50\u679c\u3092\u898b\u3066\u3044\u304f.  \n\u307e\u305a\u8fd1\u4f3c\u8aa4\u5dee\u3092\u4e00\u822c\u7684\u306a\u5f62\u3067\u5b9a\u7fa9\u3057\u3066\u304a\u304f. \n\\begin{defn}[\u8fd1\u4f3c\u8aa4\u5dee] \\\\\n  $X$\u3092\u30ce\u30eb\u30e0\u7a7a\u9593\u3068\u3059\u308b. $f \\in X$\u3068\u7a7a\u3067\u306a\u3044$Y \\subset X$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\lVert f - Y \\rVert_X := \\inf_{g \\in Y} \\lVert f - g \\rVert_X\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u4ed8\u9332\u306e\u88dc\u984c\\ref{PropOfDist}\u3088\u308a$X \\ni f \\mapsto \\lVert f - Y \\rVert_X \\in \\mathbb{R}$\u306f\u9023\u7d9a\u3067\u3042\u308b. \u3055\u3089\u306b, \u7a7a\u3067\u306a\u3044$Z \\subset X$\u306b\u5bfe\u3057\u3066, \n  \\[\n  E(Z,Y,X) := \\sup_{z \\in Z} \\lVert z - Y \\rVert_X  = \\sup_{z \\in Z} \\inf_{y \\in Y} \\lVert z - y \\rVert_X\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \n\\end{defn}\n\u6b21\u306bSobolev\u7a7a\u9593\u306e\u5b9a\u7fa9\u3092\u8ff0\u3079\u3088\u3046. Sobolev\u7a7a\u9593\u306b\u3064\u3044\u3066\u306e\u8a73\u3057\u3044\u8aac\u660e\u306f\u5bae\u5cf6\\cite{MiyajimaSobolev}\u306a\u3069\u3092\u53c2\u7167\u3055\u308c\u305f\u3044. \n\\begin{defn}[Sobolev\u7a7a\u9593] \\\\\n  \u958b\u96c6\u5408$\\Omega \\subset \\mathbb{R}^r$\u3068$p \\in [1,\\infty]$\u304a\u3088\u3073\u6574\u6570$m \\geq 1$\u306b\u5bfe\u3057\u3066, \n  \\[\n  W^{m,p}(\\Omega) \n  := \\left\\{f \\in L^p(\\Omega) ~\\vline~ \n  \\begin{aligned}\n  &|\\alpha| \\leq m \\mbox{\u306a\u308b\u4efb\u610f\u306e}\\alpha \\in \\mathbb{Z}_{\\geq 0}^r \\mbox{\u306b\u5bfe\u3057\u3066} \\\\\n  &f \\mbox{\u306e\u5f31\u5c0e\u95a2\u6570} \\partial^{\\alpha}f \\mbox{\u304c\u5b58\u5728\u3057\u3066} \\partial^{\\alpha}f \\in L^p({\\Omega})\n  \\end{aligned}\n  \\right\\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u3053\u3053\u3067, $h:\\Omega \\rightarrow \\mathbb{R}$\u304c $\\partial^{\\alpha}$\u306b\u95a2\u3059\u308b$f$\u306e\u5f31\u5c0e\u95a2\u6570\u3067\u3042\u308b\u3068\u306f, \u4efb\u610f\u306e$\\varphi \\in C_0^{\\infty}(\\Omega)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\int_{\\Omega} f(x) (\\partial^{\\alpha}\\varphi)(x) dx = (-1)^{|\\alpha|} \\int_{\\Omega} h(x)\\varphi(x)dx\n  \\]\n  \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u3044\u3046. \u3055\u3089\u306b, $f \\in W^{m,p}(\\Omega)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\lVert f \\rVert_{W^{m,p}(\\Omega)} := \\sum_{|\\alpha| \\leq m} \\lVert \\partial^{\\alpha} f \\rVert_{L^p(\\Omega)}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u307e\u305f, \n  \\[\n  \\mathcal{B}^{m,p}(\\Omega) := \\{f \\in W^{m,p}(\\Omega) \\mid \\lVert f \\rVert_{W^{m,p}(\\Omega)} \\leq 1\\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \n\\end{defn}\n\nPinkus\\cite{Pinkus}\u3067\u306f\u8fd1\u4f3c\u8aa4\u5dee$E(\\mathcal{B}^{m,p},\\mathcal{N}_r^n(\\Psi),L^p)$\u306e$n$\u306b\u95a2\u3059\u308b\u6e1b\u5c11\u7387\u306b\u3064\u3044\u3066$1999$\u5e74\u307e\u3067\u306e\u7d50\u679c\u304c\u3044\u304f\u3064\u304b\u7d39\u4ecb\u3055\u308c\u3066\u3044\u308b. \n\n\u3055\u3066, \u307e\u305a\u591a\u9805\u5f0f\u306b\u3088\u308bSobolev\u7a7a\u9593\u306e\u8fd1\u4f3c\u306b\u3064\u3044\u3066\u8ff0\u3079\u308b\u305f\u3081\u306b\u591a\u9805\u5f0f\u306e\u7a7a\u9593\u306e\u5b9a\u7fa9\u3092\u601d\u3044\u51fa\u3057\u3066\u304a\u304f. \n\\begin{defn}\n  $r,k \\in \\mathbb{N}$\u306b\u5bfe\u3057\u3066, $r$\u5909\u6570\u306e$k$\u6b21\u6589\u6b21\u591a\u9805\u5f0f\u95a2\u6570\u306e\u96c6\u5408\u3092\n  \\[\n  \\mathcal{H}_k(\\mathbb{R}^r) := \\left\\{ \\mathbb{R}^r \\ni x \\mapsto \\sum_{|\\alpha| = k} c_{\\alpha} x^{\\alpha} \\in \\mathbb{R} \\mid c_{\\alpha} \\in \\mathbb{R} \\right\\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u305f\u3060\u3057, \u548c\u306f$|\\alpha| = k$\u306a\u308b$\\alpha \\in (\\mathbb{Z}_{\\geq 0})^r$\u5168\u4f53\u306b\u308f\u305f\u308b\u3082\u306e\u3068\u3059\u308b. \u307e\u305f, \n  \\[\n  \\begin{aligned}\n  \\mathcal{P}_k(\\mathbb{R}^r) := \\bigcup_{s=0}^{k} \\mathcal{H}_s(\\mathbb{R}^r) ,~~ \\mathcal{P}(\\mathbb{R}^r) := \\bigcup_{s=0}^{\\infty} \\mathcal{H}_s(\\mathbb{R}^r)\n  \\end{aligned}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \n  \u3055\u3089\u306b$r$\u5909\u6570\u306e$k$\u6b21\u4ee5\u4e0b\u306e$\\mathbb{R}$\u4fc2\u6570\u591a\u9805\u5f0f\u5168\u4f53\u306e\u96c6\u5408\u3092\n  \\[\n  \\pi_k(\\mathbb{R}^r) := \\left\\{ \\mathbb{R}^r \\ni x \\mapsto \\sum_{|\\alpha| \\leq k} c_{\\alpha} x^{\\alpha} \\in \\mathbb{R} \\mid c_{\\alpha} \\in \\mathbb{R} \\right\\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \n\\end{defn}\n\u591a\u9805\u5f0f\u306b\u3088\u308bSobolev\u7a7a\u9593\u306e\u8fd1\u4f3c\u306b\u3064\u3044\u3066\u6b21\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u308b. \n\\begin{thm}[Jackson\u8a55\u4fa1]\\label{JacksonRate} \\\\\n  $I = (-1,1)$\u3068\u304a\u304f. $p \\in [1,\\infty], m,r \\geq 1$\u3068\u3059\u308b\u3068\u304d, \u3042\u308b$C > 0$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  E(\\mathcal{B}^{m,p}(I^r),\\pi_k(\\mathbb{R}^r), L^p(I^r)) \\leq C k^{-m} ~~~~(k=1,2,3,\\ldots)\n  \\]\n  \u3068\u306a\u308b. \n\\end{thm}\n\\begin{proof}\n  \u4ed8\u9332\u306e\u5b9a\u7406\\ref{JacksonEstimate4}\u3092\u53c2\u7167\u3055\u308c\u305f\u3044. \n\\end{proof}\n\u6b21\u306b\u6d3b\u6027\u5316\u95a2\u6570\u304c$C^{\\infty}$\u7d1a\u3067\u591a\u9805\u5f0f\u3068\u4e00\u81f4\u3057\u306a\u3044\u5834\u5408\u306e\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306b\u3088\u308b\u8fd1\u4f3c\u30ec\u30fc\u30c8\u306b\u3064\u3044\u3066\u8ff0\u3079\u308b. \u305d\u306e\u305f\u3081\u306b\u307e\u305a\u88dc\u984c\\ref{NeuralNetApproPoly}\u3092\u7cbe\u5bc6\u5316\u3057\u3088\u3046. \n\\begin{lem}\n  $n \\in \\mathbb{N}$\u3068\u3059\u308b\u3068, \u6b21\u306e\u5f0f\u304c\u6210\u308a\u7acb\u3064. \n  \\[\n  \\sum_{i=0}^n (-1)^i \\binom{n}{i} (n-2i)^j \n  = \n  \\left\\{\n  \\begin{aligned}\n  &~~0 ~~~~~~(j=0,\\ldots,n-1)\\\\\n  &~2^n n! ~~~(j=n)\n  \\end{aligned}\n  \\right.\n  \\]\n\\end{lem}\n\\begin{proof}\n  \\[\n  \\begin{aligned}\n  \\sum_{i=0}^n (-1)^i \\binom{n}{i} (n-2i)^j \n  &= \\sum_{i=0}^n (-1)^i \\binom{n}{i} \\left.\\left(\\frac{d}{dx}\\right)^j \\exp((n-2i)x)\\right|_{x=0} \\\\\n  &= \\left.\\left(\\frac{d}{dx}\\right)^j \\left(\\sum_{i=0}^n (-1)^i \\binom{n}{i} \\exp((n-2i)x) \\right)\\right|_{x=0} \\\\\n  &= \\left.\\left(\\frac{d}{dx}\\right)^j \\left(\\sum_{i=0}^n \\binom{n}{i} \\exp(x)^{n-i} (-\\exp(-x))^i \\right)\\right|_{x=0} \\\\\n  &= \\left.\\left(\\frac{d}{dx}\\right)^j \\left(\\exp(x)-\\exp(-x) \\right)^n \\right|_{x=0} \\\\\n  &= 2^n \\left.\\left(\\frac{d}{dx}\\right)^j \\left(\\sinh(x) \\right)^n \\right|_{x=0}\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u3053\u3053\u306b\n  \\[\n  \\sinh(x) = \\frac{\\exp(x)-\\exp(-x)}{2}\n  \\]\n  \u3067\u3042\u308b. \u3088\u3063\u3066\n  \\[\n  \\left.\\left(\\frac{d}{dx}\\right)^j \\left(\\sinh(x) \\right)^n \\right|_{x=0}\n  =\n  \\left\\{\n  \\begin{aligned}\n  &~~0 ~~~~(j=0,\\ldots,n-1)\\\\\n  &~~ n! ~~~(j=n)\n  \\end{aligned}\n  \\right.\n  \\]\n  \u3092\u793a\u305b\u3070\u3088\u3044. \u3053\u308c\u3092$n$\u306b\u3064\u3044\u3066\u306e\u5e30\u7d0d\u6cd5\u3067\u793a\u305d\u3046. $n=1$\u3067\u306e\u6210\u7acb\u306f\u660e\u3089\u304b. $n$\u3067\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068, $j=0,\\ldots,n-1$\u306b\u5bfe\u3057\u3066\u306f, \u7a4d\u306e\u5fae\u5206\u306b\u95a2\u3059\u308bLeibniz\u5247\u3088\u308a\n  \\[\n  \\begin{aligned}\n  &\\left.\\left(\\frac{d}{dx}\\right)^j \\left(\\sinh(x) \\right)^{n+1} \\right|_{x=0} \\\\\n  &= \\left. \\sum_{k=0}^{j} \\binom{j}{k} \\left( \\left( \\frac{d}{dx} \\right)^k (\\sinh(x))^n \\right) \\left( \\left( \\frac{d}{dx} \\right)^{j-k} \\sinh(x) \\right)  \\right|_{x=0} = 0.\n  \\end{aligned}\n  \\]\n  \u307e\u305f, \n  \\[\n  \\left.\\left(\\frac{d}{dx}\\right)^n \\left(\\sinh(x) \\right)^{n+1} \\right|_{x=0} \n  = \\left.\\left(\\left(\\frac{d}{dx}\\right)^n \\left(\\sinh(x) \\right)^n \\right) \\sinh(x) \\right|_{x=0} = 0\n  \\]\n  \u3067\u3042\u308a, \n  \\[\n  \\begin{aligned}\n  \\left.\\left(\\frac{d}{dx}\\right)^{n+1} \\left(\\sinh(x) \\right)^{n+1} \\right|_{x=0} \n  &= \\left.\\binom{n+1}{n}\\left(\\left(\\frac{d}{dx}\\right)^n \\left(\\sinh(x) \\right)^n \\right) \\left(\\frac{d}{dx} \\sinh(x) \\right)\\right|_{x=0} \\\\\n  &~~~ + \\left.\\left(\\left(\\frac{d}{dx}\\right)^{n+1} \\sinh(x)\\right) \\sinh(x) \\right|_{x=0} \\\\\n  &= (n+1)(n!) + 0 = (n+1)!\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u3053\u308c\u3067\u793a\u305b\u305f. \n\\end{proof}\n\\begin{lem}\\label{PolyApproByNNRefine}\n  $W \\subset \\mathbb{R}$\u306f$0$\u3092\u5185\u70b9\u3068\u3057\u3066\u6301\u3064\u3068\u3059\u308b. \n  \u307e\u305f, $\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f\u3042\u308b\u958b\u533a\u9593$I$\u4e0a\u3067$C^{\\infty}$\u304b\u3064\u591a\u9805\u5f0f\u3068\u4e00\u81f4\u3057\u306a\u3044\u3068\u3059\u308b. \n  \u3053\u306e\u3068\u304d, \u5404$n$\u306b\u3064\u3044\u3066\u591a\u9805\u5f0f$x \\mapsto x^n$\u306f\n  \\[\n  \\mathcal{N}_1^{n+1}(\\Psi,W,I) = \\left\\{ x \\mapsto \\sum_{i=1}^{n+1} c_i \\Psi(w_i x + b_i) \\mid c_i \\in \\mathbb{R},w_i \\in W, b_i \\in I \\right\\}\n  \\]\n  \u3067\u5e83\u7fa9\u4e00\u69d8\u8fd1\u4f3c\u3055\u308c\u308b. \u3055\u3089\u306b, $1$\u5909\u6570\u306e$n$\u6b21\u4ee5\u4e0b\u306e$\\mathbb{R}$\u4fc2\u6570\u591a\u9805\u5f0f\u306f$\\mathcal{N}_1^{2n+1}(\\Psi,W,I)$\u3067\u5e83\u7fa9\u4e00\u69d8\u8fd1\u4f3c\u3055\u308c\u308b.  \n\\end{lem}\n\\begin{proof}\n  $K \\subset \\mathbb{R}$\u3092\u7a7a\u3067\u306a\u3044\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u3068\u3059\u308b. \n  $\\Psi$\u306f$I$\u4e0a$C^{\\infty}$\u7d1a\u3067\u591a\u9805\u5f0f\u3068\u4e00\u81f4\u3057\u306a\u3044\u306e\u3067\u4ed8\u9332\u306e\u5b9a\u7406\\ref{ConditionOfSmoothFunctionIsPoly}\u3088\u308a\u3042\u308b$b \\in I$\u304c\u5b58\u5728\u3057\u3066\u4efb\u610f\u306e\u6574\u6570$n \\geq 0$\u306b\u3064\u3044\u3066$\\Psi^{(n)}(b) \\neq 0$\u3068\u306a\u308b. \n  $f:\\mathbb{R} \\times \\mathbb{R} \\ni (x,w) \\mapsto \\Psi(wx + b) \\in \\mathbb{R}$\u3068\u5b9a\u7fa9\u3059\u308b. \u524d\u534a\u3092\u793a\u3059\u306b\u306f, \u4efb\u610f\u306e$n \\in \\mathbb{N}$\u306b\u5bfe\u3057\u3066, \u95a2\u6570$x \\mapsto ((\\partial w)^n f) (x,0)$\u304c$\\mathcal{N}_1^{n+1}(\\Psi,W,I)$\u306e\u5143\u3067$K$\u4e0a\u4e00\u69d8\u8fd1\u4f3c\u3067\u304d\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u5341\u5206\u3067\u3042\u308b. \u5b9f\u969b, \u305d\u308c\u304c\u793a\u3055\u308c\u308c\u3070\n  \\[\n  x \\mapsto ((\\partial w)^n f) (x,0) = x^n \\Psi^{(n)}(0 x + b) = x^n \\Psi^{(n)}(b)\n  \\]\n  \u306a\u306e\u3067$x \\mapsto x^n$\u306f$\\mathcal{N}_1^{n+1}(\\Psi,W,I)$\u306e\u5143\u3067$K$\u4e0a\u4e00\u69d8\u8fd1\u4f3c\u3055\u308c\u308b. \u3044\u307e, $h > 0$\u306b\u5bfe\u3057\u3066, \n  \\[\n  F_x(h) := \\sum_{i=0}^n (-1)^i \\binom{n}{i} f(x,(n-2i)h) = \\sum_{i=0}^n (-1)^i \\binom{n}{i} \\Psi((n-2i)hx + b)\n  \\]\n  \u3068\u304a\u304f\u3068, Taylor\u306e\u5b9a\u7406\u3088\u308a\n  \\[\n  F_x(h) = F_x(0) + hF_x'(0)+\\cdots+\\frac{h^n}{n!}F_x^{(n)}(0)+\\frac{h^{n+1}}{(n+1)!}F_x^{(n+1)}(\\xi_x) ~~(\\exists \\xi_x \\in (0,h))\n  \\]\n  \u3068\u306a\u308b. \u524d\u547d\u984c\u3088\u308a\n  \\[\n  F_x^{(j)}(0) = ((\\partial w)^j f)(x,0) \\sum_{i=0}^n (-1)^i \\binom{n}{i} (n-2i)^j \n  =\n  \\left\\{\n  \\begin{aligned}\n  &~~~~~ 0 ~~~~~~~~~~(j=0,\\ldots,n-1)\\\\\n  &~ n! 2^n ((\\partial w)^n f)(x,0) ~~~~(j=n)\n  \\end{aligned}\n  \\right.\n  \\]\n  \u3068\u306a\u308b. \u5f93\u3063\u3066, $M_1 = \\sup \\{((\\partial w)^{n+1} f)(x,w) \\mid x \\in K, w \\in [-n,n] \\}$\u3068\u304a\u304d, \n  \\[\n  M_2 = M_1 \\left| \\sum_{i=0}^n (-1)^i \\binom{n}{i} (n-2i)^{n+1} \\right|\n  \\]\n  \u3068\u304a\u3051\u3070, \u4efb\u610f\u306e$\\varepsilon>0$\u306b\u5bfe\u3057\u3066$\\delta > 0$\u3092$\\delta < \\min\\{\\varepsilon\/M_2,1\\}$\u3068\u306a\u308b\u3088\u3046\u53d6\u308b\u3053\u3068\u3067, \u4efb\u610f\u306e$0<h<\\delta$\u3068$x \\in K$\u306b\u5bfe\u3057\u3066\n  \\[\n  \\left| \\frac{F_x(h)}{(2h)^n} - ((\\partial w)^n f)(x,0) \\right|\n  \\leq \\frac{h}{(n+1)!}|F^{(n+1)}(\\xi_x)|\n  \\leq h M_2 < \\varepsilon\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, $0<h<\\delta$\u3092\u6e80\u305f\u3057\u4e14\u3064$[-nh,nh] \\subset W$\u3092\u6e80\u305f\u3059$h$\u3092\u53d6\u308c\u3070\n  \\[\n  \\mathbb{R} \\ni x \\rightarrow \\frac{F_x(h)}{(2h)^n} = \\frac{1}{(2h)^n}\\sum_{i=0}^n (-1)^i \\binom{n}{i} \\Psi((n-2i)hx + b) \\in \\mathbb{R}\n  \\]\n  \u306f$\\mathcal{N}_1^{n+1}(\\Psi,W,I)$\u306b\u5c5e\u3057, $((\\partial w)^n f)(\\cdot,0)$\u3092$K$\u4e0a\u4e00\u69d8\u8fd1\u4f3c\u3059\u308b. \u3053\u308c\u3067\u524d\u534a\u304c\u793a\u305b\u305f. \u6b21\u306b$k \\leq n$\u306b\u5bfe\u3057\u3066$n$\u6b21\u4ee5\u4e0b\u306e\u591a\u9805\u5f0f$p(x) = a_nx^n + \\cdots + a_0$\u3092\u4efb\u610f\u306b\u53d6\u308b. \u3053\u306e\u3068\u304d, \u4e0a\u3067\u793a\u3057\u305f\u3053\u3068\u306b\u3088\u308a\u4efb\u610f\u306e$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066$h > 0$\u304c\u5b58\u5728\u3057\u3066$[-nh,nh] \\subset W$\u4e14\u3064\n  \\[\n  \\sup_{x \\in K} \\left| x^j - \\frac{\\sum_{i=0}^j (-1)^i \\binom{j}{i} \\Psi((j-2i)hx + b) }{\\Psi^{(k)}(b)(2h)^j} \\right| < \\varepsilon ~~~~(j=0,\\ldots,n)\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, \n  \\[\n  G(x) = \\sum_{j=0}^n a_j \\frac{\\sum_{i=0}^j (-1)^i \\binom{j}{i} \\Psi((j-2i)hx + b) }{\\Psi^{(k)}(b)(2h)^j}\n  \\]\n  \u3068\u304a\u304f\u3068, \n  \\[\n  \\sup_{x \\in K} |p(x)-G(x)| < \\sum_{i=0}^n |a_i| \\varepsilon\n  \\]\n  \u3068\u306a\u308b. \u305d\u3057\u3066, \n  \\[\n  S = \\{ (j-2i)h \\mid ~j =0,\\ldots,n, ~i = 0,\\ldots,j  \\}\n  \\]\n  \u306e\u8981\u7d20\u6570\u306f$2n+1$\u3067\u3042\u308b\u306e\u3067, \n  \\[\n  G(x) = \\sum_{j=1}^{2n+1} c_j \\Psi(w_jx + b) ~~~(c_j \\in \\mathbb{R}, w_j \\in W )\n  \\]\n  \u3068\u8868\u3055\u308c\u308b. \u3053\u308c\u3067\u5f8c\u534a\u3082\u793a\u305b\u305f. \n\\end{proof}\n\n\\begin{thm}\\label{NonPolyActFuncApproRate}\n  $I = (-1,1)$\u3068\u304a\u304f. $\\Psi: \\mathbb{R} \\rightarrow \\mathbb{R}$\u306f\u3042\u308b\u958b\u533a\u9593$J$\u4e0a\u3067$C^{\\infty}$\u7d1a\u304b\u3064\u591a\u9805\u5f0f\u3068\u4e00\u81f4\u3057\u306a\u3044\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u5404$p \\in [1,\\infty], m,r \\geq 1$\u306b\u5bfe\u3057\u3066\u3042\u308b$C > 0$\u304c\u5b58\u5728\u3057\u3066, \u4efb\u610f\u306e\u6574\u6570$n \\geq 1$\u306b\u5bfe\u3057\u3066\n  \\[\n  E(\\mathcal{B}^{m,p}(I^r), \\mathcal{N}_r^{n}(\\Psi),L^p(I^r)) \\leq C n^{-m\/r}\n  \\]\n  \u3068\u306a\u308b. \n\\end{thm}\n\\begin{proof}\n  $C$\u306f$r$\u306b\u4f9d\u5b58\u3057\u3066\u3088\u3044\u306e\u3067$3r \\leq n$\u3068\u3057\u3066\u3088\u3044. $\\binom{r-1+k}{k}(2k+1) \\leq n$\u306a\u308b\u6700\u5927\u306e\u6574\u6570$k \\geq 1$\u3092\u53d6\u308b. \n  \u547d\u984c\\ref{PolyRidgeRelation}\u3088\u308a, $l := \\mathrm{dim}\\mathcal{H}_k(\\mathbb{R}^r) = \\binom{r-1+k}{k}$\u3068\u304a\u304f\u3068, $a_1,\\ldots,a_l \\in \\mathbb{R}^r$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\pi_k(\\mathbb{R}^r) = \\left\\{ x \\mapsto \\sum_{i=1}^l g_i(a_i^{\\mathrm{T}} x )  \\mid g_i \\in \\pi_k(\\mathbb{R}) \\right\\}\n  \\]\n  \u3068\u306a\u308b. \u307e\u305f, \u88dc\u984c\\ref{PolyApproByNNRefine}\u3088\u308a, $\\pi_k(\\mathbb{R})$\u306e\u5143\u306f$\\mathcal{N}_1^{2k+1}(\\Psi)$\u306b\u3088\u308a\u5e83\u7fa9\u4e00\u69d8\u8fd1\u4f3c\u3055\u308c\u308b. \u3057\u305f\u304c\u3063\u3066, \n  \\[\n  \\pi_k(\\mathbb{R}^r) \\subset \\overline{\\mathcal{N}_r^{l(2k+1)}(\\Psi)}\n  \\]\n  \u3068\u306a\u308b(\u305f\u3060\u3057, \u5e83\u7fa9\u4e00\u69d8\u53ce\u675f\u306e\u4f4d\u76f8\u306b\u95a2\u3059\u308b\u9589\u5305\u3092\u3068\u3063\u3066\u3044\u308b). \u3086\u3048\u306b, Jackson\u8a55\u4fa1(\u5b9a\u7406\\ref{JacksonRate})\u3088\u308a$k$\u306b\u4f9d\u5b58\u3057\u306a\u3044\u5b9a\u6570$C > 0$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  E(\\mathcal{B}^{m,p}(I^r), \\overline{\\mathcal{N}_r^{n}(\\Psi)}, L^p(I^r)) \n  & E(\\mathcal{B}^{m,p}(I^r), \\overline{\\mathcal{N}_r^{l(2k+1)}(\\Psi)}, L^p(I^r)) \\\\\n  &\\leq E(\\mathcal{B}^{m,p}(I^r), \\pi_k(\\mathbb{R}^r), L^p(I^r)) \\leq C k^{-m}\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3053\u3053\u3067, \n  \\[\n  E(\\mathcal{B}^{m,p}(I^r), \\mathcal{N}_r^n(\\Psi), L^p(I^r)) \\leq E(\\mathcal{B}^{m,p}(I^r), \\overline{\\mathcal{N}_r^n(\\Psi)}, L^p(I^r))\n  \\]\n  \u3092\u793a\u3059. \u4efb\u610f\u306b$f \\in \\overline{\\mathcal{N}_r^n(\\Psi)}, g \\in \\mathcal{B}^{m,p}(I^r)$\u3092\u3068\u308b. $I^r$\u4e0a\u3067$f$\u306b\u4e00\u69d8\u53ce\u675f\u3059\u308b\u5217$f_j \\in \\mathcal{N}_r^n(\\Psi)$\u304c\u3068\u308c\u308b. \u3059\u308b\u3068, \n  \\[\n  \\begin{aligned}\n  \\lVert g-f  \\rVert_{L^p(I^r)} \n  &= \\lim_{j \\rightarrow \\infty} \\lVert g-f_j \\rVert_{L^p(I^r)} \\\\ &\\geq \\inf_{h \\in \\mathcal{N}_r^n(\\Psi)} \\lVert g - h \\rVert_{L^p(I^r)} = \\lVert g - \\mathcal{N}_r^n(\\Psi)\\rVert_{L^p(I^r)}\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. $f \\in \\overline{\\mathcal{N}_r^n(\\Psi)}$\u306f\u4efb\u610f\u306a\u306e\u3067, \n  \\[\n  \\lVert g - \\overline{\\mathcal{N}_r^n(\\Psi)} \\rVert_{L^p(I^r)} \\geq \\lVert g - \\mathcal{N}_r^n(\\Psi)\\rVert_{L^p(I^r)}\n  \\]\n  \u3067\u3042\u308b. \u3088\u3063\u3066, \n  \\[\n  E(\\mathcal{B}^{m,p}(I^r), \\mathcal{N}_r^n(\\Psi), L^p(I^r)) \\leq E(\\mathcal{B}^{m,p}(I^r), \\overline{\\mathcal{N}_r^n(\\Psi)}, L^p(I^r))\n  \\]\n  \u3067\u3042\u308b. \u4ee5\u4e0a\u3088\u308a, \n  \\[\n  E(\\mathcal{B}^{m,p}(I^r), \\mathcal{N}_r^n(\\Psi), L^p(I^r)) \\leq C k^{-m} \n  \\]\n  \u3068\u306a\u308b. \u3044\u307e$k$\u306e\u53d6\u308a\u65b9\u304b\u3089\n  \\[\n  n \\leq \\binom{(r-1)+(k+1)}{k+1}(2k+3)\n  \\]\n  \u3068\u306a\u308b\u304c, $k$\u306b\u4f9d\u5b58\u3057\u306a\u3044\u5b9a\u6570$C_1 > 0$\u304c\u5b58\u5728\u3057\u3066\n  \\[\n  \\begin{aligned}\n  \\binom{(r-1)+(k+1)}{k+1} \n  &= \\frac{((k+1)+(r-1))((k+1)+(r-2))\\cdots((k+1)+1)}{(r-1)(r-2) \\cdots 1} \\\\\n  &= \\left( \\frac{k+1}{r-1} + 1 \\right)\\left( \\frac{k+1}{r-2} + 1 \\right)\\cdots\\left( \\frac{k+1}{1} + 1 \\right) \\\\\n  &\\leq C_1 (k+1)^{r-1} \n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b\u306e\u3067, $k$\u306b\u4f9d\u5b58\u3057\u306a\u3044\u5b9a\u6570$C_2 > 0$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  n \\leq \\binom{r+k}{k+1}(2k+3) \\leq C_2 k^r\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, $C C_2$\u3092\u6539\u3081\u3066$C$\u3068\u304a\u3051\u3070\n  \\[\n  E(\\mathcal{B}^{m,p}(I^r), \\mathcal{N}_r^n(\\Psi), L^p(I^r)) \\leq C n^{-m\/r}\n  \\]\n  \u3067\u3042\u308b. \n\\end{proof}\n\n\u6b21\u306b\u30ea\u30c3\u30b8\u95a2\u6570\u306b\u3088\u308b\u8fd1\u4f3c\u306b\u3064\u3044\u3066\u4ee5\u4e0b\u304c\u6210\u308a\u7acb\u3064. \n\\begin{thm}\n  $I = (-1,1)$\u3068\u304a\u304f. \u307e\u305f, $p \\in [1,\\infty], m \\geq 1, r \\geq 2$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u3042\u308b$C > 0$\u304c\u5b58\u5728\u3057\u3066, \u4efb\u610f\u306e\u6574\u6570$n \\geq 1$\u306b\u5bfe\u3057\u3066\n  \\[\n  E(\\mathcal{B}^{m,p}(I^r),\\mathcal{R}_r^{n},L^p(I^r)) \\leq C n^{-m\/(r-1)}\n  \\]\n  \u304c\u6210\u308a\u7acb\u3064. \u3053\u3053\u3067\n  \\[\n  \\mathcal{R}_r^n := \\left\\{ x \\mapsto \\sum_{i=1}^n g_i(a_i^{\\mathrm{T}} x) \\mid a_i \\in \\mathbb{R}^r, g_i \\in C(\\mathbb{R}) \\right\\}\n  \\]\n  \u3067\u3042\u308b. \n\\end{thm}\n\\begin{proof} \\\\\n  $n \\geq r$\u3068\u3057\u3066\u3088\u3044. $k$\u3092$\\binom{r-1+k}{k} \\leq n$\u306a\u308b\u6700\u5927\u306e\u6574\u6570\u3068\u3059\u308b. \n  \u547d\u984c\\ref{PolyRidgeRelation}\u3088\u308a, $l := \\mathrm{dim}\\mathcal{H}_k(\\mathbb{R}^r) = \\binom{r-1+k}{k}$\u3068\u304a\u304f\u3068, $a_1,\\ldots,a_l \\in \\mathbb{R}^r$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\pi_k(\\mathbb{R}^r) = \\left\\{ x \\mapsto \\sum_{i=1}^l g_i(a_i^{\\mathrm{T}} x )  \\mid g_i \\in \\pi_k(\\mathbb{R}) \\right\\}\n  \\]\n  \u3068\u306a\u308b. \u3057\u305f\u304c\u3063\u3066, $\\pi_k(\\mathbb{R}^r) \\subset \\mathcal{R}_r^l$\u3068\u306a\u308b. \u3086\u3048\u306b, Jackson\u8a55\u4fa1(\u5b9a\u7406\\ref{JacksonRate})\u3088\u308a$k$\u306b\u4f9d\u5b58\u3057\u306a\u3044\u5b9a\u6570$C > 0$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  E(\\mathcal{B}^{m,p}(I^r),\\mathcal{R}_r^n, L^p(I^r)) \n  &\\leq E(\\mathcal{B}^{m,p}(I^r),\\mathcal{R}_r^l, L^p(I^r)) \\\\\n  \\leq E(\\mathcal{B}^{m,p}(I^r), \\pi_k(\\mathbb{R}^r), L^p(I^r)) \\leq C k^{-m}\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3044\u307e$k$\u306b\u4f9d\u5b58\u3057\u306a\u3044$C' > 0$\u304c\u3042\u3063\u3066$n \\leq C' k^{r-1}$\u3067\u3042\u308b\u304b\u3089, \n  $CC'$\u3092\u6539\u3081\u3066$C$\u3068\u304a\u3051\u3070\n  \\[\n  E(\\mathcal{B}^{m,p}(I^r),\\mathcal{R}_r^n, L^p(I^r)) \\leq C n^{-m\/(r-1)}\n  \\]\n  \u3068\u306a\u308b. \n\\end{proof}\n\u4e0a\u306e\u5b9a\u7406\u3068\u6b21\u306e\u7d50\u679c\u3092\u5408\u308f\u305b\u308b\u3053\u3068\u3067\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306b\u3088\u308b\u8fd1\u4f3c\u306b\u95a2\u3059\u308b\u7d50\u679c\u304c\u5f97\u3089\u308c\u308b. \n\\begin{thm}[Maiorov and Pinkus 1999 \\cite{MaiorovPinkus1999}] \\\\\n  \u72ed\u7fa9\u5358\u8abf\u5897\u52a0\u306asigmoidal\u95a2\u6570$\\Psi \\in C^{\\infty}(\\mathbb{R})$\u3067\u3042\u3063\u3066, \u6b21\u306e\u6027\u8cea\u3092\u307f\u305f\u3059\u3082\u306e\u304c\u5b58\u5728\u3059\u308b: \u4efb\u610f\u306e\u6574\u6570$n,r \\geq 1$\u3068$g \\in \\mathcal{R}_r^n, \\varepsilon > 0$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\sup_{x \\in B(0,1)} \\left\\lvert g(x) - \\sum_{i=1}^{r+n+1} c_i \\Psi(w_i^{\\mathrm{T}} x + b_i) \\right\\rvert < \\varepsilon\n  \\]\n  \u306a\u308b$c_i,b_i \\in \\mathbb{R}, w_i \\in \\mathbb{R}^r$\u304c\u5b58\u5728\u3059\u308b. \n\\end{thm}\n\\begin{cor}\n  $I=(-1,1)$\u3068\u304a\u304f. \n  \u72ed\u7fa9\u5358\u8abf\u5897\u52a0\u306asigmoidal\u95a2\u6570$\\Psi \\in C^{\\infty}(\\mathbb{R})$\u3067\u3042\u3063\u3066, \u6b21\u306e\u6027\u8cea\u3092\u307f\u305f\u3059\u3082\u306e\u304c\u5b58\u5728\u3059\u308b: $p \\in [1,\\infty], m \\geq 1, r \\geq 2$\u306b\u5bfe\u3057\u3066, \n  \\[\n  E(\\mathcal{B}^{m,p}(I^r), \\mathcal{N}_r^{r+n+1}(\\Psi), L^p(I^r)) \\leq C r^{-m\/(n-1)}.\n  \\]\n\\end{cor}\n\u4ee5\u4e0a\u306e\u7d50\u679c\u306f\u8fd1\u4f3c\u8aa4\u5dee\u306e\u4e0a\u304b\u3089\u306e\u8a55\u4fa1\u3067\u3042\u3063\u305f. \u4e00\u65b9\u3067\u6b21\u306e\u7d50\u679c\u306f\u8fd1\u4f3c\u8aa4\u5dee\u3092\u4e0b\u304b\u3089\u8a55\u4fa1\u3059\u308b\u3082\u306e\u3067\u3042\u308b. \n\\begin{thm}[Maiorov and Meir 1999 \\cite{MaiorovMeir2000}, Theorem5] \\\\\n  $I = (-1,1)$\u3068\u3057$p \\in [1,\\infty], m \\geq 1, r \\geq 2$\u3068\u3059\u308b. \u6d3b\u6027\u5316\u95a2\u6570$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u304c\n  \\[\n  \\Psi(t) = \\frac{1}{1+\\exp(-t)}\n  \\]\n  \u3067\u3042\u308b\u3068\u304d, \u3042\u308b\u5b9a\u6570$C > 0$\u304c\u5b58\u5728\u3057\u3066, \u4efb\u610f\u306e$n \\geq 1$\u306b\u5bfe\u3057\u3066\n  \\[\n  E(\\mathcal{B}^{m,p}(I^r), \\mathcal{N}_r^{n}(\\Psi),L^p(I^r)) \\geq C (n\\log(n))^{-m\/r}\n  \\]\n  \u304c\u6210\u308a\u7acb\u3064. \n\\end{thm}\n\n\\subsection{Barron space\u3068Barron-Maurey-Jones\u8a55\u4fa1}\n\u524d\u7bc0\u306e\u7d50\u679c\u306f, Sobolev\u7a7a\u9593\u3092\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u3067\u8fd1\u4f3c\u3059\u308b\u969b\u306e\u30ec\u30fc\u30c8\u306e\u8a55\u4fa1\u3067\u3042\u3063\u305f. \u305d\u3057\u3066,  Sobolev\u7a7a\u9593\u306e\u8fd1\u4f3c\u8aa4\u5dee\u306e\u4e0a\u304b\u3089\u306e\u8a55\u4fa1\u306b\u3064\u3044\u3066\u306f, \u6c42\u3081\u3089\u308c\u308b\u7cbe\u5ea6$\\varepsilon$\u306b\u5bfe\u3057\u3066\u4e2d\u9593\u30e6\u30cb\u30c3\u30c8\u6570$n$\u3092$\\varepsilon^{-r\/m}$\u4ee5\u4e0a\u306b\u53d6\u308b\u5fc5\u8981\u304c\u3042\u308a, \u5165\u529b\u306e\u6b21\u5143$r$\u306b\u95a2\u3057\u3066\u6307\u6570\u95a2\u6570\u7684\u306b\u5897\u5927\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u3053\u308c\u306f\u6b21\u5143\u306e\u546a\u3044\u3068\u3057\u3070\u3057\u3070\u547c\u3070\u308c\u308b\u73fe\u8c61\u3067\u3042\u308b. \n\u3067\u306f, \u9006\u306b, \u3053\u306e\u6b21\u5143\u306e\u546a\u3044\u304b\u3089\u89e3\u653e\u3055\u308c\u308b, \u3064\u307e\u308a\u76ee\u6a19\u306e\u7cbe\u5ea6\u3092\u9054\u6210\u3059\u308b\u305f\u3081\u306b\u5fc5\u8981\u306a\u4e2d\u9593\u30e6\u30cb\u30c3\u30c8\u6570\u304c$r$\u306b\u95a2\u3057\u3066\u3088\u308a\u7de9\u3084\u304b\u306b\u5897\u5927\u3059\u308b, \u3042\u308b\u3044\u306f$r$\u3068\u7121\u95a2\u4fc2\u3067\u3042\u308b\u3088\u3046\u306a\u95a2\u6570\u7a7a\u9593\u306f\u3069\u306e\u3088\u3046\u306a\u3082\u306e\u3067\u3042\u308d\u3046\u304b? \n\u3053\u306e\u3088\u3046\u306a\u95a2\u6570\u7a7a\u9593\u306fBarron space\u3068\u547c\u3070\u308c, \u73fe\u5728\u3082\u7814\u7a76\u304c\u7d9a\u3044\u3066\u3044\u308b(Weinan et al.\\cite{Weinan2019}, Siegel,Xu\\cite{Siegel2020}). \n\u672c\u7bc0\u3067\u306f, \u300cG-variation(\u5f8c\u8ff0)\u306b\u95a2\u3057\u3066\u6709\u754c\u306a\u7a7a\u9593\u300d\u304c\u3053\u306e\u554f\u984c\u306e\u3072\u3068\u3064\u306e\u7b54\u3048\u3092\u4e0e\u3048\u308b\u3053\u3068\u3092\u793a\u3059. \u3053\u308c\u306fKainen et al.\\cite{Kainen2013}\u3067\u7d39\u4ecb\u3055\u308c\u3066\u3044\u308b\u4e8b\u5b9f\u3067\u3042\u308b. \n\n\u307e\u305a, Barron-Maurey-Jones\u8a55\u4fa1\u3068\u547c\u3070\u308c\u308b\u6b21\u306e\u7d50\u679c\u304c\u3042\u308b. \n\\begin{defn}\n  $X$\u3092$\\mathbb{R}$\u4e0a\u306e\u7dda\u5f62\u7a7a\u9593\u3068\u3059\u308b\u3068\u304d, $n \\in \\mathbb{N}$\u3068\u90e8\u5206\u96c6\u5408$G \\subset X$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\mathrm{conv}_n(G) := \\left\\{\\sum_{i=1}^n a_i g_i \\mid a_i \\geq 0,~ \\sum_{i=1}^n a_i = 1,~ g_i \\in G \\right\\}\n  \\]\n  \u3068\u5b9a\u3081\u308b. \u307e\u305f, \n  \\[\n  \\mathrm{conv}(G) := \\bigcup_{n = 1}^{\\infty} \\mathrm{conv}_n(G)\n  \\]\n  \u3068\u5b9a\u3081\u308b. $\\mathrm{conv}(G)$\u306f$G$\u3092\u542b\u3080\u6700\u5c0f\u306e\u51f8\u96c6\u5408\u3067\u3042\u308b. \n\\end{defn}\n\\begin{thm}[Barron-Maurey-Jones \\cite{Kurkova2003}]\\label{MaureyRate} \\\\\n  $X$\u3092\u5185\u7a4d\u7a7a\u9593\u3068\u3059\u308b. $G \\subset X$\u3092\u7a7a\u3067\u306a\u3044\u6709\u754c\u96c6\u5408\u3068\u3057, $s_G := \\sup_{g \\in G}\\lVert g \\rVert_X$\u3068\u304a\u304f\u3068, \u4efb\u610f\u306e$f \\in \\mathrm{cl}(\\mathrm{conv}(G))$\u3068$n \\in \\mathbb{N}$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\lVert f - \\mathrm{conv}_n(G) \\rVert_X \\leq \\sqrt{\\frac{s_G^2 - \\lVert f \\rVert_X^2}{n}}\n  \\]\n  \u304c\u6210\u308a\u7acb\u3064. \n\\end{thm}\n\\begin{proof}\n  $X \\ni f \\mapsto \\lVert f - G \\rVert_X \\in \\mathbb{R}$\u306e\u9023\u7d9a\u6027\u3088\u308a, $f \\in \\mathrm{conv}(G)$\u306e\u5834\u5408\u306b\u8a3c\u660e\u3059\u308c\u3070\u5341\u5206\u3067\u3042\u308b. \u305d\u3053\u3067, \n  \\[\n  f = \\sum_{i=1}^m a_i h_i ~~~~(a_i \\geq 0, \\sum_{i=1}^m a_i = 1, h_i \\in G)\n  \\]\n  \u3068\u8868\u3059. \u3059\u308b\u3068, \u4e09\u89d2\u4e0d\u7b49\u5f0f\u3088\u308a\n  \\[\n  \\lVert f \\rVert \\leq \\sum_{i=1}^m a_i \\lVert h_i \\rVert \\leq \\sum_{i=1}^m s_G = s_G\n  \\]\n  \u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, \n  \\[\n  c := s_G^2 - \\lVert f \\rVert^2 \\geq 0\n  \\]\n  \u3067\u3042\u308b. \u3053\u3053\u3067, \u4ee5\u4e0b\u306e\u6027\u8cea\u3092\u6301\u3064\u5217$g_i \\in G$\u3092\u5e30\u7d0d\u7684\u306b\u69cb\u6210\u3057\u3088\u3046. \n  \\[\n  k \\in \\mathbb{N}, f_k := \\sum_{i=1}^k \\frac{g_i}{k} \\Rightarrow e_k^2 := \\lVert f - f_n \\rVert_X^2 \\leq \\frac{c}{k}.\n  \\]\n  \u3053\u306e\u3088\u3046\u306a\u3082\u306e\u304c\u69cb\u6210\u3067\u304d\u308c\u3070\u5b9a\u7406\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u306f\u660e\u3089\u304b\u3067\u3042\u308b. \u3055\u3066, \u3044\u307e, \n  \\[\n  \\begin{aligned}\n  \\sum_{j=1}^m a_j \\lVert f - h_j \\rVert^2 \n  &= \\sum_{j=1}^m a_j \\lVert f \\rVert^2 - 2 \\ip<{f,\\sum_{j=1}^m a_j h_j}> + \\sum_{j=1}^m a_j \\lVert h_j \\rVert^2 \\\\\n  &= \\lVert f \\rVert^2 - 2 \\lVert f \\rVert^2 + \\sum_{j=1}^m a_j \\lVert h_j \\rVert^2 \\\\\n  &\\leq - \\lVert f \\rVert^2 + \\sum_{j=1}^m a_j s_G^2 \\\\\n  &= s_G^2 - \\lVert f \\rVert^2 = c\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b\u306e\u3067, \u3042\u308b$j$\u306b\u3064\u3044\u3066$\\lVert f - h_j \\rVert^2 \\leq c$\u3067\u3042\u308b.  \u305d\u3053\u3067, $g_1 = h_j$\u3068\u304a\u304f. \u6b21\u306b$g_1,\\ldots,g_k \\in G$\u304c\u4e0a\u8a18\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3068\u3059\u308b. $g_{n+1} \\in G$\u3092\u9078\u3076\u305f\u3081\u306b$e_{n+1}^2$\u3092\u8a08\u7b97\u3059\u308b\u3068, \n  \\[\n  \\begin{aligned}\n  e_{n+1}^2 \n  &= \\lVert f - f_{n+1} \\rVert^2 = \\lVert \\frac{n}{n+1}(f-f_n) - \\frac{1}{n+1}(f-g_{n+1}) \\rVert^2 \\\\\n  &= \\frac{n^2}{(n+1)^2}e_n^2 + \\frac{2n}{(n+1)^2}\\ip<{f-f_n,f-g_{n+1}}> + \\frac{1}{(n+1)^2}\\lVert f - g_{n+1} \\rVert^2\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u4e00\u65b9, \n  \\[\n  \\begin{aligned}\n  &~~~\\sum_{j=1}^m a_j \\left( \\frac{2n}{(n+1)^2}\\ip<{f-f_n,f-h_j}> + \\frac{1}{(n+1)^2}\\lVert f - h_j \\rVert^2 \\right) \\\\\n  &= \\frac{1}{(n+1)^2} \\left( 2n \\ip<{f-f_n, \\sum_{j=1}^m a_j (f - h_j)}> + \\sum_{j=1}^m a_j \\lVert f - h_j \\rVert^2 \\right) \\\\\n  &= \\frac{1}{(n+1)^2} \\left( 2n \\ip<{f-f_n, f - f}> + \\sum_{j=1}^m a_j \\lVert f - h_j \\rVert^2 \\right) \\\\\n  &\\leq \\frac{c}{(n+1)^2}\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b\u304b\u3089, \u3042\u308b$j$\u306b\u3064\u3044\u3066\n  \\[\n  \\frac{2n}{(n+1)^2}\\ip<{f-f_n,f-h_j}> + \\frac{1}{(n+1)^2}\\lVert f - h_j \\rVert^2 \\leq \\frac{c}{(n+1)^2}\n  \\]\n  \u3067\u3042\u308b. \u305d\u3053\u3067$g_{n+1} = h_j$\u3068\u304a\u3051\u3070, \n  \\[\n  \\begin{aligned}\n  e_{n+1}^2 \n  &= \\frac{n^2}{(n+1)^2}e_n^2 + \\frac{2n}{(n+1)^2}\\ip<{f-f_n,f-g_{n+1}}> + \\frac{1}{(n+1)^2}\\lVert f - g_{n+1} \\rVert^2 \\\\\n  &\\leq \\frac{n^2}{(n+1)^2} \\frac{c}{n} + \\frac{c}{(n+1)^2} = \\frac{c}{n+1}\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3053\u308c\u3067\u793a\u305b\u305f. \n\\end{proof}\n\n\u6b21\u306b, \u3053\u306e\u7d50\u679c\u306e$L^p$\u7a7a\u9593\u306b\u304a\u3051\u308b\u985e\u4f3c\u3067\u3042\u308bDarken-Donahue-Gurvits-Sontag\u306b\u3088\u308b\u8a55\u4fa1\u3092\u8ff0\u3079\u308b. \u8a3c\u660e\u306b\u306f\u6b21\u306e\u4e0d\u7b49\u5f0f\u3092\u4f7f\u3046. \n\\begin{thm}[Clarkson\u306e\u4e0d\u7b49\u5f0f] \\\\\n  $(X,\\mathcal{M},\\mu)$\u3092\u6e2c\u5ea6\u7a7a\u9593\u3068\u3059\u308b. $p \\in (1,\\infty), q = p\/(p-1), a = \\min\\{p,q\\}$\u3068\u3059\u308b\u3068, \u4efb\u610f\u306e$f,g \\in L^p(X,\\mathcal{M},\\mu)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\lVert f+g \\rVert_{L^p(X)}^a + \\lVert f-g \\rVert_{L^p(X)}^a \\leq 2 (\\lVert f \\rVert_{L^p(X)}^a + \\lVert g \\rVert_{L^p(X)}^a).\n  \\]\n\\end{thm}\n\\begin{proof}\n  \u4ed8\u9332\u53c2\u7167. \n\\end{proof}\n\\begin{defn}\n  $X$\u3092$\\mathbb{R}$\u4e0a\u306e\u7dda\u5f62\u7a7a\u9593\u3068\u3059\u308b\u3068\u304d, $n \\in \\mathbb{N}$\u3068\u90e8\u5206\u96c6\u5408$G \\subset X$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\mathrm{span}_n(G) := \\left\\{\\sum_{i=1}^n a_i g_i \\mid a_i \\in \\mathbb{R},~ g_i \\in G \\right\\}\n  \\]\n  \u3068\u5b9a\u3081\u308b.\n\\end{defn}\n\\begin{thm}[Darken-Donahue-Gurvits-Sontag 1993 \\cite{ddgs-rarmrnn-93}]\\label{DarkenRate} \\\\\n  $(X,\\mathcal{M},\\mu)$\u3092\u6e2c\u5ea6\u7a7a\u9593\u3068\u3059\u308b. $G \\subset L^p(X,\\mathcal{M},\\mu),~ p \\in (1,\\infty)$\u3068\u3057, $f \\in \\mathrm{cl}(\\mathrm{conv}G)$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $r > 0$\u304c\u5b58\u5728\u3057\u3066, $G \\subset B(f;r) = \\{h \\in L^p(X,\\mathcal{M},\\mu) \\mid \\lVert f - h \\rVert_{L^p(X)}<r \\}$\u3068\u306a\u308b\u306a\u3089\u3070, \u4efb\u610f\u306e\u6574\u6570$n \\geq 1$\u306b\u3064\u3044\u3066, \n  \\[\n  \\lVert f - \\mathrm{span}_n G \\rVert_{L^p(X)} \\leq \\frac{2^{1\/a} r}{n^{1\/b}}\n  \\]\n  \u3068\u306a\u308b. \u3053\u3053\u3067, $q := p\/(p-1), a :=\\min(p,q), b:= \\max(p,q)$\u3067\u3042\u308b. \n\\end{thm}\n\\begin{proof}\n  $L^p(X,\\mathcal{M},\\mu) \\ni h \\mapsto \\lVert h - \\mathrm{span} G \\rVert_{L^p(X)} \\in \\mathbb{R}$\u306f\u9023\u7d9a\u3067\u3042\u308b\u304b\u3089, $f \\in \\mathrm{conv} G$\u306e\u5834\u5408\u3092\u8a3c\u660e\u3059\u308c\u3070\u5341\u5206\u3067\u3042\u308b. $f = \\sum_{j=1}^m w_j h_j$\u3068\u8868\u3059. \u3053\u306e\u3068\u304d, $G$\u306e\u5143\u306e\u5217$(g_i)$\u3067\u3042\u3063\u3066, $f_n = \\sum_{i=1}^n g_i\/n$\u3068\u304a\u304f\u3068, \n  \\[\n  e_n := \\lVert f - f_n \\rVert_{L^p(X)} \\leq \\frac{2^{1\/a} r}{n^{1\/b}}\n  \\]\n  \u3068\u306a\u308b\u3082\u306e\u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u5e30\u7d0d\u6cd5\u306b\u3088\u308a\u793a\u3059. \u307e\u305a, \u4efb\u610f\u306b$g \\in G$\u3092\u3068\u308a$g_1 = g$\u3068\u3059\u308b. \u3059\u308b\u3068, $G \\subset B(f;r)$\u304a\u3088\u3073$a>0$\u3088\u308a$\\lVert f - g_1 \\rVert_{L^p(X)} \\subset r \\leq 2^{1\/a} r$\u3067\u3042\u308b. $g_1,\\ldots,g_n \\in G$\u304c\u5f97\u3089\u308c\u3066\u3044\u308b\u3068\u3059\u308b. $f - f_n = 0$\u306a\u3089\u3070$g_i = g_n ~~(i \\geq n+1)$\u3068\u3059\u308c\u3070\u3088\u3044. $f-f_n \\neq 0$\u306a\u3089\u3070Hahn-Banach\u306e\u62e1\u5f35\u5b9a\u7406\u3088\u308a\u7dda\u5f62\u6c4e\u95a2\u6570$\\Pi_n : L^p(X,\\mathcal{M},\\mu) \\rightarrow \\mathbb{R}$\u3067$\\lVert \\Pi_n \\rVert = 1$\u304b\u3064$\\Pi_n(f-f_n) = \\lVert f-f_n \\rVert_X$\u306a\u308b\u3082\u306e\u304c\u53d6\u308c\u308b(\u4ed8\u9332\u306e\u7cfb\\ref{HahnBanachCor2}). $\\sum_{j=1}^m w_j (f-h_j) = 0$\u3067\u3042\u308b\u304b\u3089, \n  \\[\n  0 = \\Pi_n\\left( \\sum_{j=1}^m w_j (f-h_j) \\right) = \\sum_{j=1}^m w_j \\Pi_n (f-h_j)\n  \\]\n  \u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, \u3042\u308b$j$\u306b\u3064\u3044\u3066$\\Pi_n(f-h_j) \\leq 0$\u3067\u3042\u308b. \u3053\u306e\u3088\u3046\u306a$j$\u3092\u3072\u3068\u3064\u3068\u308a$g_{n+1} = h_j$\u3068\u304a\u304f.  \n  \\[\n  f_{n+1} = \\frac{1}{n+1}\\sum_{i=1}^{n+1} g_i = \\frac{n}{n+1}f_n + \\frac{1}{n+1}g_{n+1}\n  \\]\n  \u3067\u3042\u308b\u304b\u3089, Clarkson\u306e\u4e0d\u7b49\u5f0f\u3088\u308a, \n  \\[\n  \\begin{aligned}\n  (e_{n+1})^a \n  &= \\left\\lVert f-f_{n+1} \\right\\rVert_{L^p(X)}^a \n  = \\left\\lVert \\frac{n}{n+1}(f-f_n) + \\frac{1}{n+1}(f-g_{n+1}) \\right\\rVert_{L^p(X)}^a \\\\\n  &= 2 \\left( \\left\\lVert \\frac{n}{n+1}(f-f_n) \\right\\rVert_{L^p(X)}^a + \\left\\lVert \\frac{1}{n+1}(f-g_{n+1} \\right\\rVert_{L^p(X)}^a \n  \\right)\\\\\n  &~~~ - \\left\\lVert \\frac{n}{n+1}(f-f_n) - \\frac{1}{n+1}(f-g_{n+1}) \\right\\rVert_{L^p(X)}^a\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u4e00\u65b9, \u3044\u307e, $\\lVert \\Pi_n \\rVert = 1$\u304b\u3064$\\Pi_n (f-g_{n+1}) \\leq 0$\u3067\u3042\u308b\u304b\u3089, \n  \\[\n  \\begin{aligned}\n  &\\left\\lVert \\frac{n}{n+1}(f-f_n) - \\frac{1}{n+1}(f-g_{n+1}) \\right\\rVert_{L^p(X)} \\\\\n  &\\geq \\left\\lvert \\Pi_n \\left( \\frac{n}{n+1}(f-f_n) - \\frac{1}{n+1}(f-g_{n+1}) \\right) \\right\\rvert \\\\\n  &=\\left\\lvert \\frac{n}{n+1}\\lVert f-f_n \\rVert_{L^p(X)} - \\frac{1}{n+1}\\Pi_n(f-g_{n+1}) \\right\\rvert \\\\\n  &\\geq \\frac{n}{n+1}\\lVert f-f_n \\rVert_{L^p(X)}\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u3088\u3063\u3066, \u3053\u306e\u3053\u3068\u3068$g_{n+1} \\in G$\u304a\u3088\u3073\u5e30\u7d0d\u6cd5\u306e\u4eee\u5b9a\u3088\u308a, \n  \\[\n  \\begin{aligned}\n  (e_{n+1})^a \n  &\\leq 2 \\left( \\left\\lVert \\frac{n}{n+1}(f-f_n) \\right\\rVert_{L^p(X)}^a + \\left\\lVert \\frac{1}{n+1}(f-g_{n+1} \\right\\rVert_{L^p(X)}^a \n  \\right) \\\\\n  &~~~ - \\left( \\frac{n}{n+1}\\lVert f-f_n \\rVert_{L^p(X)} \\right)^a \\\\\n  &= \\left(\\frac{n}{n+1}\\right)^a \\lVert f-f_n \\rVert_{L^p(X)}^a + \\frac{2}{(n+1)^a}\\lVert f-g_{n+1} \\rVert_{L^p(X)}^a \\\\\n  &= \\left(\\frac{n}{n+1}\\right)^a (e_n)^a + \\frac{2}{(n+1)^a}\\lVert f-g_{n+1} \\rVert_{L^p(X)}^a \\\\\n  &\\leq \\left(\\frac{n}{n+1}\\right)^a \\left( \\frac{2^{1\/a} r}{n^{1\/b}} \\right)^a + \\frac{2 r^a}{(n+1)^a} = \\frac{2 r^a}{(n+1)^a}\\left( 1+\\frac{n^a}{n^{a\/b}} \\right)\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. $1\/a + 1\/b = 1$\u3088\u308a$a-a\/b = 1$\u3067\u3042\u308b\u304b\u3089, \n  \\[\n  (e_{n+1})^a \\leq \\frac{2 r^a}{(n+1)^a}( 1+n^{a-a\/b} ) = \\frac{2 r^a}{(n+1)^{a-1}} = \\frac{2 r^a}{(n+1)^{a\/b}}\n  \\]\n  \u3067\u3042\u308b. \u3053\u308c\u3067\u793a\u305b\u305f. \n\\end{proof}\n\u4ee5\u4e0a\u306e\u3075\u305f\u3064\u306e\u7d50\u679c\u306f\u3044\u305a\u308c\u3082$f \\in \\mathrm{cl}(\\mathrm{conv} G)$\u306b\u5bfe\u3059\u308b\u8a55\u4fa1\u3067\u3042\u308a, \u4e00\u822c\u306e$f \\in X$\u306b\u3064\u3044\u3066\u306f\u9069\u7528\u3055\u308c\u306a\u3044. \u305d\u3053\u3067$X$\u306e\u5143\u3059\u3079\u3066\u306b\u5bfe\u3057\u3066\u9069\u7528\u3067\u304d\u308b\u3088\u3046\u306a\u8a55\u4fa1\u3092\u4e0e\u3048\u3088\u3046. \u305d\u306e\u305f\u3081\u306b\u6b21\u306e\u6982\u5ff5\u3092\u5c0e\u5165\u3059\u308b. \n\\begin{defn}[$G$-variation] \\\\\n  $X$\u3092\u30ce\u30eb\u30e0\u7a7a\u9593\u3068\u3059\u308b. \u7a7a\u3067\u306a\u3044$G \\subset X$\u3068$f \\in X$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\lVert f \\rVert_{G} := \\inf\\{c > 0 \\mid f\/c \\in \\mathrm{cl}(\\mathrm{conv}(G \\cup -G))\\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u3053\u3053\u3067, $-G = \\{-g \\mid g \\in G\\}$\u3067\u3042\u308b. $\\lVert f \\rVert_{G}$\u3092$f$\u306e$G$-variation\u3068\u3044\u3046. \n\\end{defn}\n\\begin{rem}\n  $G \\subset X$\u3092\u7a7a\u3067\u306a\u3044\u6709\u754c\u96c6\u5408\u3068\u3057, $s_G := \\inf_{g \\in G} \\lVert g \\rVert_X$\u3068\u304a\u304f\u3068, $\\lVert f \\rVert_X \\leq s_G \\lVert f \\rVert_G$\u3068\u306a\u308b. \u5b9f\u969b, $b > 0$\u3067$f\/b \\in \\mathrm{cl}(\\mathrm{conv}(G \\cup -G))$\u306a\u308b\u3082\u306e\u3092\u4efb\u610f\u306b\u53d6\u308b\u3068, $f\/b = \\lim_{n \\rightarrow \\infty} h_n$\u306a\u308b$h_n \\in \\mathrm{conv}(G \\cup -G)$\u304c\u53d6\u308c\u308b. \u3059\u308b\u3068, $\\lVert h_n \\rVert_X \\leq s_G$\u3067\u3042\u308b\u306e\u3067$n \\rightarrow \\infty$\u3068\u3057\u3066$\\lVert f \\rVert_X \\leq b s_G $\u3068\u306a\u308b. \u3088\u3063\u3066, $b$\u306e\u4efb\u610f\u6027\u304b\u3089$\\lVert f \\rVert_X \\leq s_G \\lVert f \\rVert_G$\u3067\u3042\u308b. \n\\end{rem}\n\\begin{lem}\n  $X$\u3092$\\mathbb{R}$\u4e0a\u306e\u30ce\u30eb\u30e0\u7a7a\u9593\u3068\u3057, $C \\subset X$\u306f$0$\u3092\u542b\u3080\u51f8\u96c6\u5408\u3068\u3057, $m$\u3092$C$\u306eMinkowski\u6c4e\u95a2\u6570, \u3064\u307e\u308a$x \\in X$\u306b\u5bfe\u3057\u3066$m(x) := \\inf\\{c > 0 \\mid x\/c \\in C\\}$\u3068\u3059\u308b\u3068, \u4efb\u610f\u306e$x,y \\in X,~\\lambda \\geq 0$\u306b\u5bfe\u3057\u3066$m(\\lambda x) = \\lambda m(x)$, $m(x+y) \\leq m(x) + m(y)$\u3067\u3042\u308a, \n  \\[\n  \\{ x \\in X \\mid m(x) < 1 \\} \\subset C\n  \\]\n  \u3068\u306a\u308b. \u3055\u3089\u306b$-C = C$\u3067\u3042\u308b\u306a\u3089\u3070, $\\lambda \\in \\mathbb{R}$\u306b\u5bfe\u3057\u3066$m(\\lambda x) = |\\lambda|m(x)$\u3068\u306a\u308b. \u7279\u306b, $G$-variation\u306f\u4ee5\u4e0a\u3067\u8ff0\u3079\u305f\u6027\u8cea\u3092\u3059\u3079\u3066\u6301\u3061, \n  \\[\n  \\{f \\in X \\mid \\lVert f \\rVert_{G} < 1\\} \\subset \\mathrm{cl}(\\mathrm{conv}(G \\cup -G))\n  \\]\n  \u3068\u306a\u308b. \n\\end{lem}\n\\begin{proof}\n  \u307e\u305a$\\lambda > 0$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  \\lambda m(x) \n  &= \\inf\\{\\lambda c \\mid c > 0,  x\/c \\in C\\}  \\\\\n  &= \\inf\\{ \\lambda c \\mid (\\lambda c) > 0, \\lambda x\/(\\lambda c) \\in C \\} \\\\\n  &= \\inf\\{ b \\mid b> 0, \\lambda x\/b \\in X \\} = m(\\lambda x)\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u307e\u305f, $\\lambda = 0$\u306e\u3068\u304d\u306f$m(0)=0$\u3088\u308a$m(\\lambda x) = \\lambda m(x)$\u3067\u3042\u308b. \u305d\u3057\u3066, $m(x) = +\\infty$\u307e\u305f\u306f$m(y) = +\\infty$\u306e\u5834\u5408\u306f\u660e\u3089\u304b\u306b$m(x+y) \\leq m(x) + m(y)$\u3067\u3042\u308a, \u305d\u3046\u3067\u306a\u3044\u5834\u5408\u306f$x\/c, x\/b \\in C$\u306a\u308b\u4efb\u610f\u306e$c,b \\in (0,\\infty)$\u306b\u5bfe\u3057, $C$ \u304c\u51f8\u96c6\u5408\u3067\u3042\u308b\u3053\u3068\u304b\u3089, \n  \\[\n  \\frac{1}{c+b}(x+y)=\\frac{c}{c+b}\\frac{1}{c}x+\\frac{b}{c+b}\\frac{1}{b} y\\in C\n  \\]\n  \u3067\u3042\u308b. \u3088\u3063\u3066$m(x+y)\\leq c+b$\u3067\u3042\u308a, $c,b$ \u306e\u4efb\u610f\u6027\u3088\u308a$m(x+y)\\leq m(x)+m(y)$\u3068\u306a\u308b. \n  \u6b21\u306b$m(x) < 1$\u306a\u3089\u3070$m$\u306e\u5b9a\u7fa9\u3088\u308a$c \\in (0,1)$\u3067$x\/c \\in C$\u306a\u308b\u3082\u306e\u304c\u53d6\u308c\u308b. \u3059\u308b\u3068, $0 \\in C$\u3068$C$\u306e\u51f8\u6027\u3088\u308a, \n  \\[\n  x = c \\frac{x}{c} + (1-c) 0 \\in C\n  \\]\n  \u3067\u3042\u308b. \u6700\u5f8c\u306b, $-C=C$\u306a\u3089\u3070$\\lambda \\in \\mathbb{R}$\u3068$b > 0$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\frac{\\lambda x}{b} \\in C \\Longleftrightarrow \\frac{|\\lambda| x}{b} \\in C\n  \\]\n  \u3067\u3042\u308b\u306e\u3067$m(\\lambda x) = |\\lambda|m(x)$\u3068\u306a\u308b. \n\\end{proof}\n\\begin{rem}\n  Minkowski\u6c4e\u95a2\u6570\u3068\u3044\u3046\u8a00\u8449\u3092\u7528\u3044\u308c\u3070$G$-variation\u306f$\\mathrm{cl}(\\mathrm{conv}(G \\cup -G))$\u306eMinkowski\u6c4e\u95a2\u6570\u3067\u3042\u308b. \u305d\u3057\u3066\u4e0a\u306e\u88dc\u984c\u3088\u308a$G$-variation\u304c\u6709\u9650\u306a$X$\u306e\u5143\u5168\u4f53\u306e\u96c6\u5408\u306f$X$\u306e\u90e8\u5206\u7a7a\u9593\u3092\u306a\u3057, $G$-variation\u306f\u305d\u306e\u4e0a\u306e\u534a\u30ce\u30eb\u30e0\u3068\u306a\u308b. \u307e\u305f$G$\u304c$X$\u306e\u30ce\u30eb\u30e0\u306b\u95a2\u3057\u3066\u6709\u754c\u306a\u3089\u305d\u306e\u4e0a\u306e\u30ce\u30eb\u30e0\u3068\u306a\u308b. \n\\end{rem}\n\\begin{lem}\n  $X$\u3092\u30ce\u30eb\u30e0\u7a7a\u9593\u3068\u3057, $\\emptyset \\neq G \\subset X$\u3068\u3059\u308b. $c \\geq 0$\u306b\u5bfe\u3057\u3066$G(c) := \\{wg \\in \\mid |w| \\leq c, g \\in G\\}$\u3068\u304a\u304f\u3068, $\\lVert f \\rVert_G < \\infty$\u306a\u308b$f \\in X$\u306b\u5bfe\u3057\u3066, \n  \\[\n  f \\in \\mathrm{cl}(\\mathrm{conv}(G(\\lVert f \\rVert_{G})))\n  \\]\n  \u3068\u306a\u308b. \n\\end{lem}\n\\begin{proof}\n  \u3059\u3050\u4e0a\u306e\u88dc\u984c\u3088\u308a, \u4efb\u610f\u306e$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\frac{f}{\\lVert f \\rVert_V + \\varepsilon} \\in \\mathrm{cl}(\\mathrm{conv}(G \\cup -G))\n  \\]\n  \u3067\u3042\u308b. \u3086\u3048\u306b, $\\varepsilon \\rightarrow 0$\u3068\u3059\u308b\u3068$f\/\\lVert f \\rVert_G \\in \\mathrm{cl}(\\mathrm{conv}(G \\cup -G))$\u3067\u3042\u308b. \u3088\u3063\u3066, \n  \\[\n  f \\in \\lVert f \\rVert_G \\mathrm{cl}(\\mathrm{conv}(G \\cup -G)) = \\mathrm{cl} \\left(\\lVert f \\rVert_G \\mathrm{conv}(G \\cup -G)\\right) \\subset \\mathrm{cl} (\\mathrm{conv} G(\\lVert f \\rVert_G)).\n  \\]\n\\end{proof}\n\\begin{thm}\n  $X$\u3092\u30ce\u30eb\u30e0\u7a7a\u9593\u3068\u3059\u308b. \u307e\u305f, $G \\subset X$\u3092\u7a7a\u3067\u306a\u3044\u6709\u754c\u96c6\u5408\u3068\u3057, $s_G := \\sup_{g \\in G} \\lVert g \\rVert_X$\u3068\u304a\u304f. \u3053\u306e\u3068\u304d, $f \\in X$\u3068\u6574\u6570$n \\geq 1$\u306b\u5bfe\u3057\u3066, \n  \\begin{itemize}\n      \\item[(1)] $X$\u304c\u5185\u7a4d\u7a7a\u9593\u306a\u3089\u3070, \n        \\[\n        \\lVert f - \\mathrm{span}_n G \\rVert_X \\leq \\sqrt{\\frac{s_G^2 \\lVert f \\rVert_G^2 - \\lVert f \\rVert_X^2}{n}}\n        \\]\n        \u3068\u306a\u308b. \u7279\u306b, $B_R(\\lVert \\cdot \\rVert_G) := \\{g \\in X \\mid \\mid \\lVert g \\rVert_G \\leq R\\}$\u306b\u5bfe\u3057\u3066, \n        \\[\n        E(B_R(\\lVert \\cdot \\rVert_G), \\mathrm{span}_n G, X) \\leq s_G \\frac{R}{n^{1\/2}}.\n        \\]\n      \\item[(2)] \u6e2c\u5ea6\u7a7a\u9593$(Y,\\mathcal{N},\\nu)$\u3068$p \\in (1,\\infty)$\u306b\u5bfe\u3057\u3066$X = L^p(Y,\\mathcal{N},\\nu)$\u3067\u3042\u308b\u3068\u304d, \n        \\[\n        \\lVert f - \\mathrm{span}_n G \\rVert_X \\leq \\frac{2^{1+1\/a} s_G \\lVert f \\rVert_G}{n^{1\/b}}\n        \\]\n        \u3068\u306a\u308b. \u3053\u3053\u3067, $a := \\min(p,p\/(p-1)), b:= \\max(p,p\/(p-1))$\u3067\u3042\u308b. \u7279\u306b, \n        \\[\n        E(B_R(\\lVert \\cdot \\rVert_G), \\mathrm{span}_n G, L^p(Y,\\nu)) \\leq 2^{1+1\/a} s_G \\frac{R}{n^{1\/b}}.\n        \\]\n  \\end{itemize}\n\\end{thm}\n\\begin{proof}\n  $c \\geq 0$\u306b\u5bfe\u3057\u3066$G(c) := \\{wg \\in \\mid |w| \\leq c,~ g \\in G\\}$\u3068\u304a\u304f. \u3053\u306e\u3068\u304d, $\\mathrm{conv}_n G(c) \\subset \\mathrm{span}_n G(c) = \\mathrm{span}_n G$\u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, $f \\in X$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\lVert f - \\mathrm{span}_n G \\rVert_X \\leq \\lVert f - \\mathrm{conv}_n G(\\lVert f \\rVert_G ) \\rVert_X\n  \\]\n  \u3067\u3042\u308b. \u4e00\u65b9, \u3059\u3050\u4e0a\u306e\u88dc\u984c\u3088\u308a, $\\lVert f \\rVert_G < \\infty$\u306e\u3068\u304d$f \\in \\mathrm{cl}(\\mathrm{conv}(G(\\lVert f \\rVert_{G})))$\u3067\u3042\u308b. \u307e\u305f, $\\lVert f \\rVert_X \\leq s_G \\lVert f \\rVert_G$\u3088\u308a$G(\\lVert f \\rVert_G ) \\subset B(f,2s_G \\lVert f \\rVert_G)$\u3067\u3042\u308b. \u3088\u3063\u3066, \u5b9a\u7406\\ref{MaureyRate}\u3068\u5b9a\u7406\\ref{DarkenRate}\u3088\u308a\u4e3b\u5f35\u306f\u6210\u308a\u7acb\u3064. \n\\end{proof}\n\u4ee5\u4e0a\u306e\u7d50\u679c\u306f\u76f4\u63a5\u7684\u306b\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u306b\u3088\u308b\u8fd1\u4f3c\u30ec\u30fc\u30c8\u3092\u8ff0\u3079\u305f\u3082\u306e\u3067\u306f\u306a\u3044\u304c, \u6d3b\u6027\u5316\u95a2\u6570$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u304a\u3088\u3073$W \\subset \\mathbb{R}^r, B \\subset \\mathbb{R}$\u306b\u5bfe\u3057\u3066$G = G_r(\\Psi,W,B) := \\{ x \\mapsto \\Psi(w^{\\mathrm{T}} x + b) \\mid w \\in W, b \\in B \\}$\u3068\u304a\u304f\u3068\u304d, $\\Psi, W, B$\u3084\u554f\u984c\u3068\u3059\u308b\u95a2\u6570\u7a7a\u9593\u306b\u9069\u5f53\u306a\u6761\u4ef6\u3092\u52a0\u3048\u308c\u3070\u5b9a\u7406\u306e\u4eee\u5b9a($G$\u304c\u6709\u754c)\u3092\u307f\u305f\u3059\u3088\u3046\u306b\u3067\u304d\u308b. \u307e\u305f, $b_j \\in B,c_j \\in \\mathbb{R}, w_j \\in W$\u306b\u5bfe\u3057\u3066, \n\\[\nf(x) = \\sum_{j=1}^m c_j \\Psi(w_j^{\\mathrm{T}} x + b_j) ~~~(x \\in \\mathbb{R}^r)\n\\]\n\u3068\u304a\u304f\u3068, \n\\[\n\\begin{aligned}\nf(x) \n&= \\sum_{c_j > 0} c_j \\Psi(w_j^{\\mathrm{T}} x + b_j) + \\sum_{c_j \\leq 0} c_j \\Psi(w_j^{\\mathrm{T}} x + b_j) \\\\\n&= \\sum_{c_j > 0} |c_j| \\Psi(w_j^{\\mathrm{T}} x + b_j) + \\sum_{c_j \\leq 0} |c_j| (-\\Psi(w_j^{\\mathrm{T}} x + b_j))\n\\end{aligned}\n\\]\n\u3067\u3042\u308b\u306e\u3067, \n\\[\n\\lVert f \\rVert_G = \\inf\\{c > 0 \\mid f\/c \\in \\mathrm{cl}(\\mathrm{conv}(G \\cup -G))\\} \\leq \\sum_{j=1}^m |c_j|\n\\]\n\u3067\u3042\u308b\u3053\u3068\u3092\u6ce8\u610f\u3057\u3066\u304a\u304f. \n\n\u6b21\u306b\u8fd1\u4f3c\u8aa4\u5dee\u306e\u4e0b\u9650\u306b\u95a2\u3059\u308b\u5b9a\u7406\u3092\u7d39\u4ecb\u3057\u3088\u3046. \n\\begin{thm}[Makovoz 1996 \\cite{Makovoz1996} theorem4] \\\\\n  sigmoidal\u95a2\u6570$\\Psi:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f$\\Psi = \\chi_{[0,\\infty)}$\u3067\u3042\u308b\u304b, Lipschitz\u9023\u7d9a\u4e14\u3064\u3042\u308b$\\eta,C>0$\u304c\u5b58\u5728\u3057\u3066\n  \\[\n  |\\Psi(t) - \\chi_{[0,\\infty)}(t)| \\leq C|t|^{-\\eta} ~~~~(\\forall t \\in \\mathbb{R}\\setminus\\{0\\})\n  \\]\n  \u3068\u306a\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $G := G_r(\\Psi,\\mathbb{R}^r,\\mathbb{R})$\u3068\u304a\u3051\u3070, \n  \u958b\u51f8\u96c6\u5408$D \\subset \\mathbb{R}^r$\u3068$\\xi > 0$\u306b\u5bfe\u3057\u3066\u5b9a\u6570$C(D,\\xi) > 0$\u304c\u5b58\u5728\u3057\u3066, \u4efb\u610f\u306e\u6574\u6570$n \\geq 1$\u306b\u5bfe\u3057\u3066\n  \\[\n  E(B_1(\\lVert \\cdot \\rVert_G),\\mathrm{span}_n G,L^2(D)) \\geq C(D,\\xi) n^{-1\/2 - 1\/r - \\xi} .\n  \\]\n\\end{thm}\n\n\u3055\u3066, \u4ee5\u4e0a\u306e\u7d50\u679c\u306f\u51aa\u30ec\u30fc\u30c8\u306e\u7d50\u679c\u3067\u3042\u3063\u305f. \u6700\u5f8c\u306b\u6307\u6570\u30ec\u30fc\u30c8\u306e\u7d50\u679c\u3092\u8ff0\u3079\u3088\u3046. \n\\begin{thm}[K\\r{u}rkov\\'{a}-Sanguineti 2008 \\cite{KurkovaSanguineti2008}] \\\\\n  $X$\u3092Hilbert\u7a7a\u9593\u3068\u3059\u308b. \u307e\u305f, $G \\subset X$\u3092\u7a7a\u3067\u306a\u3044\u6709\u754c\u96c6\u5408\u3068\u3057, $s_G = \\sup_{g\\in G} \\lVert g \\rVert_X$\u3068\u304a\u304f. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$f \\in \\mathrm{conv} G$\u306b\u5bfe\u3057\u3066, $\\tau_f \\in [0,1)$\u304c\u5b58\u5728\u3057\u3066, \u4efb\u610f\u306e\u6574\u6570$n \\geq 1$\u306b\u3064\u3044\u3066\n  \\[\n  \\lVert f - \\mathrm{conv}_n G \\rVert_X \\leq \\sqrt{\\tau_f^{n-1}(s_G^2 - \\lVert f \\rVert_X^2)}\n  \\]\n  \u304c\u6210\u308a\u7acb\u3064. \n\\end{thm}\n\\begin{proof}\n  $\\sum_{j=1}^m a_j = 1, ~a_j > 0, ~g_j \\in G$\u306b\u3088\u308a$f = \\sum_{j=1}^m a_j g_j$\u3068\u8868\u3059. $m=1$\u306a\u3089$\\tau_f = 0$\u3068\u3057\u3066\u6210\u7acb\u3059\u308b\u306e\u3067$m > 1$\u3068\u3057\u3066\u3088\u3044. \u307e\u305f, \u3042\u308b$j$\u306b\u3064\u3044\u3066$f = g_j$\u3068\u306a\u308b\u5834\u5408\u3082\u9664\u3044\u3066\u3088\u3044. $G' := \\{g_1,\\ldots,g_m\\}$\u3068\u304a\u304f. \u5404$n = 1,\\ldots,m$\u306b\u5bfe\u3057\u3066$f_n \\in \\mathrm{conv}_n G', \\rho_n \\in (0,1)$\u304c\u5b58\u5728\u3057\u3066\n  \\[\n  \\lVert f - f_n \\rVert_X^2 \\leq (1-\\rho_n^2)^{n-1}(s_G^2 - \\lVert f \\rVert_X^2)\n  \\]\n  \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092$n$\u306b\u95a2\u3059\u308b\u5e30\u7d0d\u6cd5\u3067\u793a\u3059. \u307e\u305a, $g_{j_1} \\in G'$\u3092\n  \\[\n  \\lVert f - g_{j_1} \\rVert_X = \\min_{g \\in G'} \\lVert f - g \\rVert_X\n  \\]\n  \u306a\u308b\u3082\u306e\u3068\u3057$f = g_{j_1}$\u3068\u304a\u3051\u3070, \n  \\[\n  \\begin{aligned}\n  \\lVert f-f_1 \\rVert_X^2 \n  &= \\sum_{j=1}^m a_j \\lVert f-f_1 \\rVert_X^2 \\leq \\sum_{j=1}^m a_j \\lVert f-g_j \\rVert_X^2 \\\\\n  &= \\lVert f \\rVert_X^2 - 2 \\ip<{f,\\sum_{j=1}^m a_j g_j}> + \\sum_{j=1}^m a_j \\lVert g_j \\rVert_X^2 \\\\\n  &= \\lVert f \\rVert_X^2 - 2 \\lVert f \\rVert_X^2 + \\sum_{j=1}^m a_j \\lVert g_j \\rVert_X^2 \\\\\n  &\\leq s_G^2 - \\lVert f \\rVert_X^2\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, $\\rho_1 \\in (0,1)$\u3092\u4efb\u610f\u306b\u53d6\u308c\u3070$n=1$\u306e\u5834\u5408\u306f\u6210\u308a\u7acb\u3064. \u6b21\u306b$n-1$\u3067\u6210\u308a\u7acb\u3064\u3068\u3059\u308b. $f_{n-1}=f$\u306a\u3089\u3070$f_n = f_{n-1}, \\rho_n = \\rho_{n-1}$\u3068\u53d6\u308c\u3070\u3088\u3044\u306e\u3067$f_{n-1} \\neq f$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \n  \\[\n  \\sum_{j=1}^m a_j \\ip<{f-f_{n-1},f-g_j}> = \\ip<{f-f_{n-1},f-\\sum_{j=1}^m a_j g_j}> = \\ip<{f-f_{n-1},f-f}> = 0\n  \\]\n  \u3067\u3042\u308b\u306e\u3067, \u6b21\u306e\u3044\u305a\u308c\u304b\u304c\u6210\u308a\u7acb\u3064($m>1$\u306b\u6ce8\u610f). \n  \\begin{itemize}\n      \\item[(1)] $\\exists g \\in G' , \\ip<{f-f_{n-1},f-g}> < 0.$\n      \\item[(2)] $\\forall g \\in G', \\ip<{f-f_{n-1},f-g}> < 0.$\n  \\end{itemize}\n  (2)\u306f\u3042\u308a\u5f97\u306a\u3044\u3053\u3068\u3092\u793a\u305d\u3046. $f_{n-1} \\in \\mathrm{conv}_{n-1} G'$\u3067\u3042\u308b\u306e\u3067, \n  \\[\n  f_{n-1} = \\sum_{k=1}^{n-1} b_k g_{j_k} ~~~~(\\sum_{k=1}^{n-1}b_k=1,~b_k > 0, g_{j_1} \\in G')\n  \\]\n  \u3068\u8868\u3055\u308c\u308b. \u5f93\u3063\u3066, (2)\u304c\u6210\u308a\u7acb\u3064\u306a\u3089\u3070, \n  \\[\n  \\begin{aligned}\n  \\lVert f - f_{n-1} \\rVert_X^2\n  &= \\ip<{f-f_{n-1},f-\\sum_{k=1}^{n-1}b_k g_{j_k}}> \\\\\n  &= \\sum_{k=1}^{n-1} b_k \\ip<{f-f_{n-1},f-g_{j_k}}> = 0\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b\u306e\u3067$f=f_{n-1}$\u3068\u306a\u308b. \u3053\u308c\u306f$f \\neq f_{n-1}$\u306b\u53cd\u3059\u308b. \u3088\u3063\u3066, (2)\u306f\u3042\u308a\u5f97\u306a\u3044. \u3086\u3048\u306b(1)\u304c\u6210\u308a\u7acb\u3064. \u305d\u3053\u3067$g_{j_n} \\in G'$\u3092$\\ip<{f-f_{n-1},f-g_{j_n}}> < 0$\u3092\u6e80\u305f\u3059\u3082\u306e\u3068\u3059\u308b. \u305d\u3057\u3066, $f \\neq f_{n-1}, g_{j_n}$\u306b\u6ce8\u610f\u3057\u3066\n  \\[\n  \\rho_n := \\min \\left\\{\\rho_{n-1},- \\frac{\\ip<{f-f_{n-1},f-g_{j_n}}>}{\\lVert f - f_{n-1} \\rVert_X \\lVert f - g_{j_n} \\rVert_X} \\right\\} \\in (0,1)\n  \\]\n  \u3068\u304a\u304f. \u307e\u305f, $\\alpha_n \\in [0,1]$\u306b\u5bfe\u3057\u3066\n  \\[\n  f_n = \\alpha_n f_{n-1} + (1-\\alpha_n)g_{j_n} \\in \\mathrm{conv}_n G'\n  \\]\n  \u3068\u304a\u304f. \u4ee5\u4e0b, $\\alpha_n$\u3092\u3046\u307e\u304f\u53d6\u308b\u3053\u3068\u3067$f_n,\\rho_n$\u304c\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u3067\u304d\u308b\u3053\u3068\u3092\u793a\u3059. $q_n = -\\ip<{f-f_{n-1},f-g_{j_n}}> > 0$, $r_n =\\lVert f-g_{j_n} \\rVert_X > 0$\u3068\u304a\u304d, \u5404$k = 1,\\ldots,n$\u306b\u3064\u3044\u3066$e_k := \\lVert f - f_k \\rVert_X$\u3068\u304a\u304f\u3068, \n  \\[\n  \\begin{aligned}\n  e_n^2 \n  &= \\lVert \\alpha_n (f-f_{n-1}) + (1-\\alpha_n)(f-g_{j_n}) \\rVert_X^2 \\\\\n  &= \\alpha_n^2 e_{n-1}^2 + 2\\alpha_n(1-\\alpha_n)\\ip<{f-f_{n-1},f-g_{j_n}}> + (1-\\alpha_n)^2 \\lVert f-g_{j_n} \\rVert_X^2 \\\\\n  &= \\alpha_n^2 e_{n-1}^2 - 2\\alpha_n(1-\\alpha_n)q_n + (1-\\alpha_n)^2 r_n^2 \\\\\n  &=\\alpha_n^2(e_{n-1}^2 + 2 q_n + r_n^2) - 2\\alpha_n(q_n + r_n^2) + r_n^2\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u5f93\u3063\u3066, \n  \\[\n  \\frac{d e_n^2}{d\\alpha_n} = 2\\alpha_n(e_{n-1}^2 + 2 q_n + r_n^2) - 2(q_n + r_n^2)\n  \\]\n  \u3067\u3042\u308b. \u3088\u3063\u3066, $e_{n-1}^2 + 2 q_n + r_n^2 > 0$\u3088\u308a$e_n^2$\u306f\n  \\[\n  \\alpha_n = \\frac{q_n + r_n^2}{e_{n-1}^2 + 2 q_n + r_n^2} \\in [0,1]\n  \\]\n  \u3067\u6700\u5c0f\u5024\u3092\u3068\u308b. \u3053\u306e$\\alpha_n$\u304c\u6c42\u3081\u308b\u3082\u306e\u3067\u3042\u308b. \u5b9f\u969b, \u305d\u306e\u3068\u304d\n  \\[\n  \\begin{aligned}\n  e_n^2 \n  &= \\frac{(q_n + r_n^2)^2}{e_{n-1}^2 + 2 q_n + r_n^2} -2\\frac{(q_n + r_n^2)(q_n + r_n^2)}{e_{n-1}^2 + 2 q_n + r_n^2} + r_n^2 \\\\\n  &= \\frac{-q_n^2 -2q_n r_n^2 - r_n^4 + e_{n-1}^2 r_n^2 + 2 q_n r_n^2 + r_n^4}{e_{n-1}^2 + 2 q_n + r_n^2} \\\\\n  &= \\frac{-q_n^2 + e_{n-1}^2 r_n^2}{e_{n-1}^2 + 2 q_n + r_n^2}\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308a, $\\rho_n = \\min \\left\\{\\rho_{n-1}, q_n\/(e_{n-1}r_n) \\right\\}\n  $\u3088\u308a$\\rho_n e_{n-1} r_n \\leq q_n$\u3067\u3042\u308b\u306e\u3067, \n  \\[\n  \\begin{aligned}\n  e_n^2 \n  &\\leq \\frac{-\\rho_n^2 e_{n-1}^2 r_n^2 + e_{n-1}^2 r_n^2}{e_{n-1}^2 + 2 q_n + r_n^2} < \\frac{(1-\\rho_n^2) e_{n-1}^2 r_n^2}{r_n^2} = (1-\\rho_n^2) e_{n-1}^2 \\\\\n  &\\leq (1-\\rho_n^2) (1-\\rho_{n-1}^2)^{n-2} (s_G^2 - \\lVert f \\rVert_X^2) \\leq (1-\\rho_n^2)^{n-1} (s_G^2 - \\lVert f \\rVert_X^2)\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b(\u6700\u5f8c\u306e\u4e0d\u7b49\u53f7\u306f$\\rho_n \\leq \\rho_{n-1}$\u306b\u3088\u308b). \u3088\u3063\u3066$n$\u3067\u6210\u7acb\u3059\u308b. \\\\\n  \u3055\u3066, $\\rho_f := \\min\\{\\rho_n \\mid n=1,\\ldots,m\\}$\u3068\u304a\u304f. \u3059\u308b\u3068, $n=1,\\ldots,m$\u306b\u5bfe\u3057\u3066\n  \\[\n  \\lVert f - \\mathrm{conv}_n G \\rVert_X^2 \\leq \\lVert f - f_n \\rVert_X^2 \\leq (1-\\rho_n)^{n-1} (s_G^2 - \\lVert f \\rVert_X^2) \\leq (1-\\rho_f)^{n-1} (s_G^2 - \\lVert f \\rVert_X^2)\n  \\]\n  \u3068\u306a\u308a, $n > m$\u306b\u5bfe\u3057\u3066\u306f$\\lVert f - \\mathrm{conv}_n G \\rVert_X = 0$\u3068\u306a\u308b\u306e\u3067$\\tau_f = 1- \\rho_f^2$\u304c\u6c42\u3081\u308b\u5b9a\u6570\u3067\u3042\u308b. \n\\end{proof}\n\n\\section{\u307e\u3068\u3081\u3068\u4eca\u5f8c\u306e\u8ab2\u984c}\n\\subsubsection*{\u307e\u3068\u3081}\n\u6d3b\u6027\u5316\u95a2\u6570\u304c\u975e\u591a\u9805\u5f0f\u304b\u3064\u5c40\u6240\u6709\u754c\u3067Riemann\u7a4d\u5206\u53ef\u80fd\u306e\u5834\u5408\u306bfeedforward\u578b\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u304c\u4e07\u80fd\u8fd1\u4f3c\u80fd\u529b\u3092\u6301\u3064\u306a\u3069, \u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406\u306b\u3064\u3044\u3066, \u8907\u6570\u306e\u7d50\u679c\u3092\u8a18\u53f7\u3092\u7d71\u4e00\u3057\u3066\u89e3\u8aac\u3057\u305f. \n\u307e\u305f, \u305d\u306e\u8fd1\u4f3c\u30ec\u30fc\u30c8\u306e\u554f\u984c\u306b\u304a\u3044\u3066, \u8fd1\u4f3c\u3059\u308b\u7a7a\u9593\u3068\u3057\u3066Sobolev\u7a7a\u9593\u3092\u53d6\u3063\u305f\u5834\u5408\u3092\u6271\u3063\u305f. \n\u3055\u3089\u306b, \u305d\u306e\u5834\u5408\u306b\u306f\u6b21\u5143\u306e\u546a\u3044\u3068\u3044\u3046, \u6240\u671b\u306e\u7cbe\u5ea6\u3092\u9054\u6210\u3059\u308b\u305f\u3081\u306b\u5fc5\u8981\u306a\u4e2d\u9593\u30e6\u30cb\u30c3\u30c8\u6570\u304c\u5165\u529b\u306e\u6b21\u5143\u306b\u95a2\u3057\u3066\u6307\u6570\u95a2\u6570\u7684\u306b\u5897\u5927\u3059\u308b\u73fe\u8c61\u304c\u767a\u751f\u3059\u308b\u3053\u3068\u3092\u8aac\u660e\u3057\u305f. \n\u305d\u3057\u3066, \u6b21\u5143\u306e\u546a\u3044\u304b\u3089\u89e3\u653e\u3055\u308c\u308b\u7a7a\u9593\u3068\u3057\u3066Barron space\u306e\u6982\u5ff5\u3092\u5c0e\u5165\u3057, Barron space\u306e\u3072\u3068\u3064\u3068\u3057\u3066\u300cG-variation\u306b\u95a2\u3057\u3066\u6709\u754c\u306a\u7a7a\u9593\u300d\u3092\u7d39\u4ecb\u3057\u305f. \n\n\\subsubsection*{\u4eca\u5f8c\u306e\u8ab2\u984c}\n\u672c\u8ad6\u6587\u3067\u89e6\u308c\u308b\u3053\u3068\u304c\u3067\u304d\u306a\u304b\u3063\u305f\u3053\u3068\u306b, Deep narrow network\u306e\u4e07\u80fd\u8fd1\u4f3c\u5b9a\u7406\\cite{DeepNarrowNetworks}\u3084\u6df1\u5c64\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u8fd1\u4f3c\u30ec\u30fc\u30c8(Yarotsky 2016 \\cite{Yarotsky}\u306a\u3069)\u304c\u3042\u308b. \n\u307e\u305f, Barron space\u306b\u95a2\u3057\u3066\u306f, Barron\u81ea\u8eab\u306e\u7d50\u679c\u3084\u3054\u304f\u6700\u8fd1\u306eWeinan et al.\\cite{Weinan2019},  Siegel, Xu\\cite{Siegel2020}\u306e\u7d50\u679c\u3092\u307e\u3068\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u306a\u304b\u3063\u305f. \n\u4eca\u5f8c\u306f, \u3053\u3046\u3057\u305f\u6df1\u5c64\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3084Barron space\u306e\u8fd1\u5e74\u306e\u9032\u5c55\u3092\u307e\u3068\u3081\u305f\u3044. \n\u3055\u3089\u306b, \u8fd1\u5e74\u6ce8\u76ee\u3092\u96c6\u3081\u3066\u3044\u308b, \u65e5\u672c\u306e\u7814\u7a76\u8005\u304c\u5275\u59cb\u3057\u305f\u30cb\u30e5\u30fc\u30e9\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u7a4d\u5206\u8868\u73fe\u7406\u8ad6(Murata 1996 \\cite{NoboruMurata}\u3084Sonoda, Murata 2015 \\cite{SonodaMurata}\u306a\u3069)\u306e\u89e3\u8aac\u3092\u8a66\u307f\u305f\u3044. \n\n\n\\section*{\u8b1d\u8f9e}\n\u672c\u8ad6\u6587\u306e\u4f5c\u6210\u3092\u652f\u63f4\u3057\u3066\u304f\u3060\u3055\u3063\u305f\u4e5d\u5dde\u5927\u5b66\u30de\u30b9\u30fb\u30d5\u30a9\u30a2\u30fb\u30a4\u30f3\u30c0\u30b9\u30c8\u30ea\u7814\u7a76\u6240\u306e\u6e9d\u53e3\u4f73\u5bdb\u6559\u6388\u306b\u6df1\u304f\u611f\u8b1d\u3044\u305f\u3057\u307e\u3059. \n\u307e\u305f, \u672c\u8ad6\u6587\u306e\u3044\u304f\u3064\u304b\u306e\u547d\u984c\u306e\u8a3c\u660e\u306e\u5b8c\u6210\u306b\u3054\u5354\u529b\u9802\u3044\u305f, \n\u7247\u5ca1\u4f51\u592a\u3055\u3093, \u4e5d\u5dde\u5927\u5b66\u5927\u5b66\u9662\u6570\u7406\u5b66\u7814\u7a76\u9662\u306e\u9ad8\u7530\u4e86\u51c6\u6559\u6388, \u6176\u61c9\u7fa9\u587e\u5927\u5b66\u7d4c\u6e08\u5b66\u90e8\u306e\u670d\u90e8\u54f2\u5f25\u6559\u6388\u306b\u539a\u304f\u5fa1\u793c\u7533\u3057\u4e0a\u3052\u307e\u3059. \n\u6700\u5f8c\u306b\u306a\u308a\u307e\u3059\u304c, \u5bb6\u65cf\u3084\u53cb\u4eba, \u4e26\u3073\u306b\u8ad6\u6587\u4f5c\u6210\u3092\u3054\u652f\u63f4\u3044\u305f\u3060\u3044\u305f\u5168\u3066\u306e\u65b9\u3005\u306b\u611f\u8b1d\u306e\u610f\u3092\u8868\u3057\u307e\u3059. \n\n\n\\section{\u4ed8\u9332}\n\n\\subsection{\u6e2c\u5ea6\u306e\u57fa\u672c\u6027\u8cea}\n\u672c\u7bc0\u3067\u306f, \u6e2c\u5ea6\u306e\u57fa\u672c\u6027\u8cea\u3092\u8a3c\u660e\u3059\u308b. \u307e\u305a, \u53ef\u6e2c\u7a7a\u9593, \u6e2c\u5ea6\u306e\u5b9a\u7fa9\u3092\u8ff0\u3079\u3088\u3046. \n\\begin{defn}[\u53ef\u6e2c\u7a7a\u9593] \\\\\n  \u96c6\u5408$X$\u3068$X$\u306e\u90e8\u5206\u96c6\u5408\u7cfb$\\mathcal{M}$\u306e\u7d44$(X, \\mathcal{M})$\u304c\u53ef\u6e2c\u7a7a\u9593\u3067\u3042\u308b\u3068\u306f, \n  \\begin{itemize}\n      \\item[(1)] $\\emptyset \\in \\mathcal{M}$.\n      \\item[(2)] $\\forall E \\in \\mathcal{M}, X \\setminus E \\in \\mathcal{M}$.\n      \\item[(3)] \u4efb\u610f\u306e$\\mathcal{M}$\u306e\u5143\u306e\u5217$(E_n)_{n\\in \\mathbb{N}}$\u306b\u3064\u3044\u3066$\\bigcup_{n \\in\\mathbb{N}} E_n \\in \\mathcal{M}$.\n  \\end{itemize}\n  \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u3044\u3046. \n\\end{defn}\n\\begin{rem}\n  $(X, \\mathcal{M})$\u3092\u53ef\u6e2c\u7a7a\u9593\u3068\u3059\u308b\u3068,  $\\mathcal{M}$\u306f\u53ef\u7b97\u500b\u307e\u3067\u306e\u5171\u901a\u90e8\u5206\u3092\u3068\u308b\u64cd\u4f5c\u306b\u95a2\u3057\u3066\u9589\u3058\u3066\u3044\u308b. \u5b9f\u969b, $(E_n)_{n\\in \\mathbb{N}}$\u3092$\\mathcal{M}$\u306e\u5143\u306e\u5217\u3068\u3059\u308b\u3068, \n  \\[\n  \\bigcup_{n=1}^{\\infty} E_n = X \\setminus \\left(\\bigcup_{n=1}^{\\infty} (X \\setminus E_n) \\right) \\in \\mathcal{M}.\n  \\]\n\\end{rem}\n\\begin{defn}[\u6e2c\u5ea6, \u6e2c\u5ea6\u7a7a\u9593] \\\\\n  $(X, \\mathcal{M})$\u3092\u53ef\u6e2c\u7a7a\u9593\u3068\u3059\u308b\u3068\u304d, \u5199\u50cf$\\mu:\\mathcal{M} \\rightarrow [0,\\infty]$\u304c$(X, \\mathcal{M})$\u4e0a\u306e\u6e2c\u5ea6\u3067\u3042\u308b\u3068\u306f, \n  \\begin{itemize}\n      \\item[(1)] $\\mu(\\emptyset) = 0$.\n      \\item[(2)] $\\mathcal{M}$\u306e\u5143\u306e\u5217$(E_n)_{n\\in \\mathbb{N}}$\u304c$E_n \\cap E_m = \\emptyset ~(n \\neq m)$\u3092\u6e80\u305f\u3059\u306a\u3089\u3070, \n      \\[\n      \\mu\\left( \\bigcup_{n\\in \\mathbb{N}} E_n \\right) = \\sum_{n=1}^{\\infty} \\mu(E_n).\n      \\]\n  \\end{itemize}\n  \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u3044\u3046. \u3053\u306e\u3068\u304d, \u4e09\u3064\u7d44$(X,\\mathcal{M},\\mu)$\u3092\u6e2c\u5ea6\u7a7a\u9593\u3068\u3044\u3046. \n\\end{defn}\n\\begin{lem}\n  $(X, \\mathcal{M}, \\mu)$\u3092\u6e2c\u5ea6\u7a7a\u9593\u3068\u3059\u308b\u3068\u304d, \u4ee5\u4e0b\u304c\u6210\u308a\u7acb\u3064. \n  \\begin{itemize}\n      \\item[(1)]$A,B \\in \\mathcal{M}$\u304c$B \\subset A$\u3092\u6e80\u305f\u3059\u3068\u304d, $\\mu(B) \\leq \\mu(A).$\n      \\item[(2)]$A,B \\in \\mathcal{M}$\u304c$B \\subset A$\u3092\u6e80\u305f\u3059\u3068\u304d$\\mu(B) < \\infty$\u306a\u3089\u3070$\\mu(A \\setminus B ) = \\mu(A) - \\mu(B).$\n      \\item[(3)]$\\mathcal{M}$\u306e\u5143\u306e\u5217$(A_n)_{n=1}^{\\infty}$\u306b\u5bfe\u3057\u3066, \n      \\[\n      \\mu\\left( \\bigcup_{n=1}^{\\infty} A_n \\right) \\leq \\sum_{n=1}^{\\infty} \\mu(A_n).\n      \\]\n      \\item[(4)]$\\mathcal{M}$\u306e\u5143\u306e\u5217$(A_n)_{n=1}^{\\infty}$\u304c$A_n \\subset A_{n+1} ~~(\\forall n)$\u3092\u6e80\u305f\u3059\u306a\u3089\u3070, \n      \\[\n      \\lim_{n \\rightarrow \\infty} \\mu(A_n) =  \\mu\\left(\\bigcup_{n=1}^{\\infty} A_n \\right).\n      \\]\n      \\item[(5)]$\\mathcal{M}$\u306e\u5143\u306e\u5217$(A_n)_{n=1}^{\\infty}$\u304c$A_n \\supset A_{n+1} ~~(\\forall n)$\u3092\u6e80\u305f\u3059\u3068\u304d$\\mu(A_1) < \\infty$\u306a\u3089\u3070\n      \\[\n      \\lim_{n \\rightarrow \\infty} \\mu(A_n) =  \\mu\\left(\\bigcap_{n=1}^{\\infty} A_n \\right).\n      \\]\n  \\end{itemize}\n  $(1)$\u3092\u6e2c\u5ea6\u306e\u5358\u8abf\u6027, $(3)$\u3092\u6e2c\u5ea6\u306e\u52a3\u52a0\u6cd5\u6027, $(4),(5)$\u3092\u6e2c\u5ea6\u306e\u9023\u7d9a\u6027\u3068\u3044\u3046. \n\\end{lem}\n\\begin{proof} \\\\\n  $(1)$: $\\mu(A) = \\mu((A \\setminus B) \\cup B) = \\mu(A \\setminus B) + \\mu(B) \\geq \\mu(B).$\\\\\n  $(2)$: $(1)$\u306e\u8a3c\u660e\u306b\u304a\u3044\u3066$\\mu(B)$\u3092\u4e21\u8fba\u304b\u3089\u5f15\u3051\u3070\u3088\u3044. \\\\\n  $(3)$: $B_1 = A_1$\u3068\u3057, $B_{n+1} = A_{n+1} \\setminus \\bigcup_{j=1}^{n} A_j$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \n  \\[\n  \\bigcup_{n=1}^{\\infty} B_n = \\bigcup_{n=1}^{\\infty} A_n\n  \\]\n  \u3067\u3042\u308a, $n \\neq m$\u306a\u3089\u3070$B_n \\cap B_m = \\emptyset$\u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, \n  \\[\n  \\begin{aligned}\n  \\mu\\left( \\bigcup_{n=1}^{\\infty} A_n \\right) \n  &= \\mu\\left( \\bigcup_{n=1}^{\\infty} B_n \\right) \\\\\n  &= \\sum_{n=1}^{\\infty} \\mu(B_n) \\\\\n  &\\leq \\sum_{n=1}^{\\infty} \\mu(A_n) ~~~(\\because (1)).\n  \\end{aligned}\n  \\]\n  \\\\\n  $(4)$:$A_0 = \\emptyset$\u3068\u3057, $B_n := A_n - A_{n-1}$\u3068\u304a\u304f. \u3059\u308b\u3068, $B_ \\in \\mathcal{M}$\u3067, $n \\neq m$\u306b\u5bfe\u3057\u3066$B_n \\cap B_m = \\emptyset$\u3068\u306a\u308b. \u3055\u3089\u306b, $\\bigcup_{n=1}^{\\infty} A_n = \\bigcup_{n=1}^{\\infty} B_n$\u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, \u6e2c\u5ea6\u306e\u5b8c\u5168\u52a0\u6cd5\u6027\u3088\u308a, \n  \\[\n  \\begin{aligned}\n  \\mu(\\bigcup_{n=1}^{\\infty} A_n) \n  &= \\mu(\\bigcup_{n=1}^{\\infty} B_n) \\\\\n  &= \\sum_{n=1}^{\\infty} \\mu(B_n) \\\\\n  &= \\sum_{n=1}^{\\infty} (\\mu(A_n) - \\mu(A_{n-1})) ~~~(\\because (2)) \\\\\n  &= \\lim_{n \\rightarrow \\infty} (\\mu(A_n) - \\mu(A_0)) \\\\\n  &= \\lim_{n \\rightarrow \\infty} \\mu(A_n). \n  \\end{aligned}\n  \\]\n  \\\\\n  $(5)$: $(\\mu(A_n))_{n=1}^{\\infty}$\u306f\u975e\u8ca0\u306a\u5358\u8abf\u6e1b\u5c11\u5217\u306a\u306e\u3067\u6975\u9650\u304c\u5b58\u5728\u3059\u308b\u3053\u3068, \u307e\u305f, $\\mu(A_1) < \\infty$\u3088\u308a$\\mu( \\bigcap_{j=1}^{\\infty} A_j) \\leq \\mu(A_n) < \\infty ~(\\forall n)$\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b. \n  \u3044\u307e, $C_n := X \\setminus A_n$\u3068\u304a\u304f\u3068, $C_n \\in \\mathcal{M}$\u3067\u3042\u308a, $C_n \\subset C_{n+1}~(\\forall n)$\u3067\u3042\u308b\u304b\u3089, \n  \\[\n  \\begin{aligned}\n  \\mu(X) - \\mu\\left( \\bigcap_{n=1}^{\\infty} A_n \\right)\n  &=\\mu\\left( X \\setminus \\bigcap_{n=1}^{\\infty} A_n \\right) ~~~(\\because (2))\\\\\n  &= \\mu\\left(\\bigcup_{n=1}^{\\infty} X \\setminus A_n \\right) \\\\\n  &= \\lim_{n \\rightarrow \\infty} \\mu(X \\setminus A_n) ~~~(\\because (4))\\\\\n  &= \\lim_{n \\rightarrow \\infty} (\\mu(X) - \\mu(A_n)) ~~~(\\because (2))\\\\\n  &= \\mu(X) - \\lim_{n \\rightarrow \\infty} \\mu(A_n)\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, \u6c42\u3081\u308b\u7b49\u5f0f\u3092\u5f97\u308b. \n\\end{proof}\n\n\\subsection{\u5358\u95a2\u6570\u8fd1\u4f3c\u5b9a\u7406}\n\n\u672c\u7bc0\u3067\u306f\u975e\u8ca0\u5024\u53ef\u6e2c\u95a2\u6570\u304c\u5358\u8abf\u5897\u52a0\u306a\u975e\u8ca0\u5358\u95a2\u6570\u5217\u3067\u8fd1\u4f3c\u3067\u304d\u308b\u3053\u3068\u3092\u793a\u3059. \u3053\u3053\u3067\u53ef\u6e2c\u7a7a\u9593$(X,\\mathcal{M})$\u4e0a\u306e\u5358\u95a2\u6570\u3068\u306f, \n\\[\nf = \\sum_{j=1}^m c_j \\chi_{E_j} ~~~~(m \\in \\mathbb{N}, c_j \\in \\mathbb{R}, E_j \\in \\mathcal{M} )\n\\]\n\u3068\u8868\u3055\u308c\u308b\u95a2\u6570\u306e\u3053\u3068\u3092\u3044\u3046. \u305f\u3060\u3057$\\chi_E$\u3067$E$\u306e\u5b9a\u7fa9\u95a2\u6570\u3092\u8868\u3059. \n\\begin{thm}[\u5358\u95a2\u6570\u8fd1\u4f3c\u5b9a\u7406] \\\\\n  $f \\geq 0$\u3092\u53ef\u6e2c\u7a7a\u9593$(X,\\mathcal{M})$\u4e0a\u306e\u53ef\u6e2c\u95a2\u6570\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $n = 1,2,\\ldots$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  &A_k^n := f^{-1}([(k-1)\/2^n, k\/2^n)) ~~(k=1,\\ldots,2^n n), \\\\\n  &A_0^n := f^{-1}([n,\\infty)), \\\\\n  &\\varphi_n := \\sum_{k=1}^{2^n n} \\frac{k-1}{2^n} \\chi_{A_k^n} + n\\chi_{A_0^n} \n  \\end{aligned}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b\u3068, \u4efb\u610f\u306e$x \\in X$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  &0 \\leq \\varphi_1(x) \\leq \\varphi_2(x) \\leq \\cdots \\leq \\varphi_n(x) \\leq \\cdots \\leq f(x),  \\\\\n  &f(x) = \\lim_{n \\rightarrow \\infty} \\varphi_n(x)\n  \\end{aligned}\n  \\]\n  \u304c\u6210\u308a\u7acb\u3064. \n\\end{thm}\n\\begin{proof}\n  \u4efb\u610f\u306b$x \\in X$\u3092\u53d6\u308b. \u307e\u305f, \u4efb\u610f\u306b$n$\u3092\u53d6\u308b. \u3053\u306e\u3068\u304d, $\\varphi_n(x) \\leq \\varphi_{n+1}(x)$\u3092\u793a\u305d\u3046. \u307e\u305a$f(x) \\geq n+1$\u306e\u5834\u5408\u306f, $\\varphi_n(x)=n \\leq n+1 = \\varphi_{n+1}(x)$\u3067\u6210\u7acb\u3059\u308b. \u6b21\u306b$f(x) < n$\u306e\u3068\u304d\u306f, $f(x) \\in [(k-1)\/2^n, k\/2^n)$\u306a\u308b$k \\in \\{1,\\ldots, 2^n n\\}$\u304c\u53d6\u308c\u308b. \u3059\u308b\u3068, \n  \\[\n  \\frac{2k-2)}{2^{n+1}} \\leq f(x) \\leq \\frac{2k-1}{2^{n+1}}\n  \\]\n  \u307e\u305f\u306f, \n  \\[\n  \\frac{2k-1)}{2^{n+1}} \\leq f(x) \\leq \\frac{2k}{2^{n+1}}\n  \\]  \n  \u3067\u3042\u308b. \u524d\u8005\u306a\u3089$\\varphi_n(x) = (k-1)\/2^n = 2(k-1)\/2^{n+1} = \\varphi_{n+1}(x)$\u3067\u6210\u7acb\u3059\u308b. \u5f8c\u8005\u306a\u3089$\\varphi_n(x) = (k-1)\/2^n = 2(k-1)\/2^{n+1} \\leq (2k-1)\/2^{n+1}$\u3067\u6210\u7acb\u3059\u308b. \u6700\u5f8c\u306b$n \\leq f(x) < n+1$\u306e\u5834\u5408\u306f, $(k-1)\/2^{n+1} \\leq f(x) < k\/2^{n+1}$\u306a\u308b$k \\in \\{1+2^{n+1}n, \\ldots, 2^{n+1}(n+1)\\}$\u304c\u53d6\u308c\u308b. \u3057\u305f\u304c\u3063\u3066, \n  \\[\n  \\varphi_n(x) = n \\leq (k-1)\/2^{n+1} = \\varphi_{n+1}(x)\n  \\]\n  \u3068\u306a\u3063\u3066\u6210\u7acb\u3059\u308b. \u6b21\u306b$\\varphi_n(x) \\rightarrow f(x)$\u3092\u793a\u3059. \u4efb\u610f\u306e$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, $f(x) > N$\u304b\u3064$1\/2^{N} < \\varepsilon$\u306a\u308b$N$\u304c\u53d6\u308c\u308b. \u3053\u306e\u3068\u304d, $n \\geq n$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\frac{k-1}{2^n} \\leq f(x) < \\frac{k}{2^n}\n  \\]\n  \u306a\u308b$k \\in \\{1,\\ldots, 2^n n\\}$\u304c\u3068\u308c, $\\varphi_n(x) = (k-1)\/2^n$\u3068\u306a\u308b\u306e\u3067, \n  \\[\n  0 \\leq f(x) - \\varphi_n(x) \\leq \\frac{1}{2^n} < \\frac{1}{2^N} < \\varepsilon\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066$\\varphi_n(x) \\rightarrow f(x)$\u3067\u3042\u308b. \u540c\u6642\u306b$\\varphi_n(x) \\leq f(x)$\u3082\u793a\u3055\u308c\u305f. \n\\end{proof}\n\n\\subsection{Lebesgue\u7a4d\u5206\u306e\u57fa\u672c\u5b9a\u7406}\n\u672c\u7bc0\u3067\u306fLebesgue\u7a4d\u5206\u8ad6\u306e\u57fa\u672c\u5b9a\u7406\u3092\u8aac\u660e\u3059\u308b. \u8a73\u3057\u3044\u89e3\u8aac\u3084\u8a3c\u660e\u306f\u732a\u72e9\\cite{igarizitukaiseki}\u306a\u3069\u3092\u53c2\u7167\u3055\u308c\u305f\u3044.  \n\u307e\u305a\u7a4d\u5206\u306e\u5b9a\u7fa9\u3092\u8ff0\u3079\u308b. \n\\begin{defn}\n  $(X,\\mathcal{M},\\mu)$\u3092\u6e2c\u5ea6\u7a7a\u9593\u3068\u3059\u308b. \u975e\u8ca0\u5358\u95a2\u6570$f:X \\rightarrow [0,\\infty)$, \u3064\u307e\u308a$a_1,\\ldots,a_k \\geq 0$\u3068$E_1,\\ldots,E_k \\in \\mathcal{M}$\u306b\u3088\u308a$f = \\sum_{j=1}^k a_j \\chi_{E_j}$\u3068\u8868\u3055\u308c\u308b$f$\u306b\u5bfe\u3057\u3066\u306f\n  \\[\n  \\int_X f d\\mu = \\int_X f(x) d\\mu(x) := \\sum_{j=1}^k a_j \\mu(E_j)\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u975e\u8ca0\u306e\u53ef\u6e2c\u95a2\u6570$f:X \\rightarrow [0,\\infty)$\u306b\u5bfe\u3057\u3066\u306f\n  \\[\n  \\int_X f(x) d\\mu(x) := \\sup\\left\\{ \\int_X \\varphi d\\mu ~\\vline~ 0 \\leq \\varphi \\leq f , \\varphi\\mbox{\u306f\u5358\u95a2\u6570} \\right\\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u53ef\u6e2c\u95a2\u6570$f:X \\rightarrow \\mathbb{R}$\u306b\u5bfe\u3057\u3066\u306f$\\int_X |f(x)| d\\mu(x) < \\infty$\u3067\u3042\u308b\u5834\u5408\u306b\n  \\[\n  \\int_X f(x) d\\mu(x) := \\int_X f^{+}(x) d\\mu(x) - \\int_X f^{-}(x) d\\mu(x)\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u3053\u3053\u3067$f^{+} = \\max(0,f),~ f^{-} = \\max(0,-f)$\u3067\u3042\u308b. \n\\end{defn}\n\u6b21\u306b\u53ce\u675f\u5b9a\u7406\u306b\u3064\u3044\u3066\u8ff0\u3079\u308b. \n\\begin{thm}[\u5358\u8abf\u53ce\u675f\u5b9a\u7406] \\\\\n  $(\\Omega, \\mathcal{F}, \\mu)$\u3092\u6e2c\u5ea6\u7a7a\u9593\u3068\u3059\u308b. $(f_n)$\u306f$(\\Omega, \\mathcal{F}, \\mu)$\u4e0a\u306e\u975e\u8ca0\u5024\u53ef\u6e2c\u95a2\u6570\u5217\u3067\u95a2\u6570$f:\\Omega \\rightarrow \\mathbb{R}$\u306b\u5bfe\u3057\u3066\n  \\[\n  f_1(x) \\leq f_2(x)\\leq \\cdots \\leq f_n(x) \\rightarrow f(x) ~~~~(\\forall x \\in \\Omega)\n  \\]\n  \u3092\u6e80\u305f\u3059\u3068\u3059\u308b\u3068, \n  \\[\n  \\lim_{n \\rightarrow \\infty}{\\int_{\\Omega}{f_n}{d\\mu}} = \\int_{\\Omega}{f}{d\\mu}\n  \\]\n  \u304c\u6210\u308a\u7acb\u3064. \n\\end{thm}\n\\begin{thm}[\u512a\u53ce\u675f\u5b9a\u7406] \\\\\n$(\\Omega, \\mathcal{F}, \\mu)$\u3092\u6e2c\u5ea6\u7a7a\u9593\u3068\u3059\u308b. $(f_n)$\u304c$(\\Omega, \\mathcal{F}, \\mu)$\u4e0a\u306e\u53ef\u6e2c\u95a2\u6570\u5217\u3067, \u95a2\u6570 $f$\u306b\u5404\u70b9\u53ce\u675f\u3057, \u304b\u3064\u3042\u308b$(\\Omega, \\mathcal{F}, \\mu)$\u4e0a\u306e\u53ef\u7a4d\u5206\u95a2\u6570$g$\u304c\u5b58\u5728\u3057\u3066\u4efb\u610f\u306e$n, x$\u306b\u5bfe\u3057\u3066$\\left\\|f_n(x)\\right\\| \\leq g(x)$\u304c\u6210\u308a\u7acb\u3064\u306a\u3089\u3070, $f$\u306f\u53ef\u7a4d\u5206\u3067, \\[\n\\lim_{n \\rightarrow \\infty}{\\int_{\\Omega}{f_n}{d\\mu}} = \\int_{\\Omega}{f}{d\\mu}\n\\]\n\u304c\u6210\u308a\u7acb\u3064. \n\\end{thm}\n\\begin{thm}[\u7a4d\u5206\u8a18\u53f7\u4e0b\u306e\u5fae\u5206]\\label{DiffinIntegral} \\\\\n  $(\\Omega, \\mathcal{F}, \\mu)$\u3092\u6e2c\u5ea6\u7a7a\u9593\u3068\u3057$a < b$\u3068\u3059\u308b. $f:\\Omega \\times (a,b) \\rightarrow \\mathbb{R}$\u306f\u6b21\u306e\u4e8c\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3068\u3059\u308b. \n  \\begin{itemize}\n      \\item[(1)] \u5404$t \\in (a,b)$\u306b\u3064\u3044\u3066$f(\\cdot,t):\\Omega \\ni x \\mapsto f(t,x) \\in R$\u304c$L^1(\\Omega, \\mathcal{F}, \\mu)$\u306b\u5c5e\u3057, \u5404\u70b9$(x,t)$\u306b\u304a\u3044\u3066$(\\partial_t f)(x,t)$\u304c\u5b58\u5728\u3059\u308b. \n      \\item[(2)] $0 \\leq g \\in L^1(\\Omega, \\mathcal{F}, \\mu)$\u304c\u5b58\u5728\u3057\u3066$|(\\partial_t f)(x,t)| \\leq g(x) ~~~(\\forall x \\in \\Omega,t \\in (a,b)).$\n  \\end{itemize}\n  \u3053\u306e\u3068\u304d, $F(t) := \\int_{\\Omega} f(x,t) d\\mu(x)$\u306f$(a,b)$\u4e0a\u3067\u5fae\u5206\u53ef\u80fd\u3067\u3042\u3063\u3066, \n  \\[\n  F'(t) = \\int_{\\Omega} \\frac{\\partial f}{\\partial t}(x,t) d \\mu(x).\n  \\]\n\\end{thm}\n\\begin{proof}\n  $t_0 \\in (a,b)$\u3092\u56fa\u5b9a\u3057$t_0$\u306b\u53ce\u675f\u3059\u308b\u6570\u5217$t_n \\in (a,b), ~t_n \\neq t_0$\u3092\u53d6\u308a, $x \\in \\Omega$\u306b\u5bfe\u3057\u3066\n  \\[\n  h_n(x) := \\frac{f(x,t_n) - f(x,t_0)}{t_n - t_0} ~~~~(n=1,2,\\ldots,)\n  \\]\n  \u3068\u304a\u304f. \u3053\u306e\u3068\u304d\u5404$x$\u306b\u3064\u3044\u3066$(\\partial_t f )(x,t_0) = \\lim_{n \\rightarrow \\infty} h_n(x)$\u3067\u3042\u308a, \u5e73\u5747\u5024\u306e\u5b9a\u7406\u3088\u308a\n  \\[\n  |h_n(x)| \\leq \\sup_{t \\in (a,b)} \\left| \\frac{\\partial f}{\\partial t}(x,t) \\right| \\leq g(x)\n  \\]\n  \u3067\u3042\u308b. \u3088\u3063\u3066, \u512a\u53ce\u675f\u5b9a\u7406\u3088\u308a\n  \\[\n  \\lim_{n \\rightarrow \\infty} \\frac{F(t_n)-F(t_0)}{t_n-t_0} = \\lim_{n \\rightarrow \\infty} \\int_{\\Omega} h_n(x) d\\mu(x) = \\int_{\\Omega} \\frac{\\partial f}{\\partial t}(x,t_0) d\\mu(x).\n  \\]\n  \u3086\u3048\u306b\u6570\u5217$(t_n)$\u306e\u4efb\u610f\u6027\u304b\u3089\u5b9a\u7406\u306f\u793a\u3055\u308c\u305f. \n\\end{proof}\n\u6700\u5f8c\u306bFubini\u306e\u5b9a\u7406\u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u3088\u3046. \n\\begin{defn}[$\\sigma$-\u6709\u9650\u6e2c\u5ea6\u7a7a\u9593] \\\\\n \u6e2c\u5ea6\u7a7a\u9593$(X,\\mathcal{M},\\mu)$\u304c$\\sigma$-\u6709\u9650\u3067\u3042\u308b\u3068\u306f, $E_n \\in \\mathcal{M} ~~(n=1,2,\\ldots)$\u304c\u5b58\u5728\u3057\u3066$\\mu(E_n) < \\infty$\u4e14\u3064$X = \\bigcup_{n \\in \\mathbb{N}} E_n$\u3068\u306a\u308b\u3053\u3068\u3092\u3044\u3046. \n\\end{defn}\n\\begin{defn}[\u76f4\u7a4d\u53ef\u6e2c\u7a7a\u9593] \\\\\n  $(X,\\mathcal{M}),(Y,\\mathcal{N})$\u3092\u53ef\u6e2c\u7a7a\u9593\u3068\u3059\u308b. $X \\times Y$\u4e0a\u306e$\\sigma$-\u52a0\u6cd5\u65cf$\\mathcal{M} \\otimes \\mathcal{N}$\u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u3081\u308b. \\[\n  \\mathcal{M} \\otimes \\mathcal{N} := \\sigma\\left[ \\{E \\times F \\mid E \\in \\mathcal{M}, F \\in \\mathcal{N}\\} \\right]\n  \\]\n  \u3053\u306e\u3068\u304d, \u53ef\u6e2c\u7a7a\u9593$(X \\times Y, \\mathcal{M} \\otimes \\mathcal{N})$\u3092$(X,\\mathcal{M}),(Y,\\mathcal{N})$\u306e\u76f4\u7a4d\u53ef\u6e2c\u7a7a\u9593\u3068\u3044\u3046. \n\\end{defn}\n\\begin{defn}[\u76f4\u7a4d\u6e2c\u5ea6\u7a7a\u9593] \\\\\n  $(X,\\mathcal{M},\\mu),(Y,\\mathcal{N},\\nu)$\u3092$\\sigma$-\u6709\u9650\u306a\u6e2c\u5ea6\u7a7a\u9593\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u76f4\u7a4d\u53ef\u6e2c\u7a7a\u9593$(X \\times Y, \\mathcal{M} \\otimes \\mathcal{N})$\u4e0a\u306e\u6e2c\u5ea6$\\mu \\otimes \\nu$\u3067\n  \\[\n  (\\mu \\otimes \\nu)(E \\times F) = \\mu(E)\\nu(F) ~~~~(E \\in \\mathcal{M}, F \\in \\mathcal{N})\n  \\]\n  \u3092\u6e80\u305f\u3059\u3082\u306e\u304c\u4e00\u610f\u7684\u306b\u5b58\u5728\u3059\u308b. $\\mu \\otimes \\nu$\u3092$\\mu,\\nu$\u306e\u76f4\u7a4d\u6e2c\u5ea6\u3068\u3044\u3044, \u6e2c\u5ea6\u7a7a\u9593$(X \\times Y, \\mathcal{M} \\otimes \\mathcal{N}, \\mu \\otimes \\nu)$\u3092$(X,\\mathcal{M},\\mu),(Y,\\mathcal{N},\\nu)$\u306e\u6e2c\u5ea6\u7a7a\u9593\u3068\u3044\u3046. \n\\end{defn}\n\\begin{thm}[Tonelli\u306e\u5b9a\u7406] \\\\\n  $(X,\\mathcal{M},\\mu),(Y,\\mathcal{N},\\nu)$\u3092$\\sigma$-\u6709\u9650\u306a\u6e2c\u5ea6\u7a7a\u9593\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $(X \\times Y, \\mathcal{M} \\otimes \\mathcal{N}$\u4e0a\u306e\u53ef\u6e2c\u95a2\u6570$f \\geq 0$\u306b\u5bfe\u3057\u3066\u6b21\u304c\u6210\u308a\u7acb\u3064. \n  \\begin{itemize}\n      \\item[(1)] $F_1(x) := \\int_{Y} f(x,y) d\\nu(y)$\u306f$X$\u4e0a\u306e\u53ef\u6e2c\u95a2\u6570\u3067\u3042\u308a, \\\\\n      $F_2(y) := \\int_{X} f(x,y) d\\mu(x)$\u306f$Y$\u4e0a\u306e\u53ef\u6e2c\u95a2\u6570\u3067\u3042\u308b. \n      \\item[(2)] \u3055\u3089\u306b, \n      \\[\n      \\int_{X \\times Y} f(x,y) d(\\mu \\otimes \\nu)(x,y) = \\int_{X} F_1(x) d\\mu(x) = \\int_{Y} F_2(y) d\\nu(y).\n      \\]\n  \\end{itemize}\n\\end{thm}\n\\begin{thm}[Fubini\u306e\u5b9a\u7406] \\\\\n  $(X,\\mathcal{M},\\mu),(Y,\\mathcal{N},\\nu)$\u3092$\\sigma$-\u6709\u9650\u306a\u6e2c\u5ea6\u7a7a\u9593\u3068\u3059\u308b. $f \\in L^1(X \\times Y, \\mathcal{M} \\otimes \\mathcal{N}, \\mu \\otimes \\nu)$\u306b\u5bfe\u3057\u3066\u6b21\u304c\u6210\u308a\u7acb\u3064. \n  \\begin{itemize}\n      \\item[(1)] $\\mu$-a.e.$x$\u3067$f(x,\\cdot) \\in L^1(Y,\\mathcal{N},\\nu)$\u4e14\u3064$\\nu$-a.e.$y$\u3067$f(\\cdot,y) \\in L^1(X,\\mathcal{M},\\mu)$.\n      \\item[(2)] $\\mu$-a.e.$x$\u3067\u5b9a\u7fa9\u3055\u308c\u308b$F_1(x) := \\int_{Y} f(x,y) d\\nu(y)$\u306b\u3064\u3044\u3066$F \\in L^1(X,\\mathcal{M},\\mu)$. \u307e\u305f, $\\nu$-a.e.$y$\u306b\u5bfe\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b$F_2(y) := \\int_{X} f(x,y) d\\mu(x)$\u306b\u3064\u3044\u3066$F_2 \\in L^1(Y,\\mathcal{N},\\nu)$\u3068\u306a\u308b. \n      \\item[(3)] \u3055\u3089\u306b, \n      \\[\n      \\int_{X \\times Y} f(x,y) d(\\mu \\otimes \\nu)(x,y) = \\int_{X} F_1(x) d\\mu(x) = \\int_{Y} F_2(y) d\\nu(y).\n      \\]\n  \\end{itemize}\n\\end{thm}\n\n\\subsection{Hahn-Banach\u306e\u62e1\u5f35\u5b9a\u7406}\n\u672c\u7bc0\u3067\u306fHahn-Banach\u306e\u5b9a\u7406\u306e\u4e3b\u5f35\u306e\u8aac\u660e\u3092\u3057, \u672c\u8ad6\u6587\u3067\u7528\u3044\u305f\u3044\u304f\u3064\u304b\u306e\u4e8b\u5b9f\u3092Hahn-Banach\u306e\u5b9a\u7406\u304b\u3089\u5c0e\u51fa\u3059\u308b. \n\\begin{defn}[\u52a3\u7dda\u5f62\u6c4e\u95a2\u6570] \\\\\n$V$\u3092$\\mathbb{R}$\u4e0a\u306e\u7dda\u5f62\u7a7a\u9593\u3068\u3059\u308b. \u5199\u50cf$p : V \\rightarrow \\mathbb{R}$\u304c\u52a3\u7dda\u5f62\u3067\u3042\u308b\u3068\u306f, \u4ee5\u4e0b\u306e2\u3064\u306e\u6761\u4ef6\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a00\u3046:\n\\[\n\\begin{aligned}\n&(1)~~\\forall \\lambda > 0, \\forall x \\in V, ~~ p(\\lambda x) = \\lambda p(x) \\\\\n&(2)~~\\forall x, y \\in V, ~~ p(x+y) \\leq p(x)+p(y)\n\\end{aligned}\n\\]\n\\end{defn}\n\n\u4f8b\u3048\u3070, \u30ce\u30eb\u30e0\u7a7a\u9593\u306b\u304a\u3044\u3066\u30ce\u30eb\u30e0\u306f\u52a3\u7dda\u5f62\u6c4e\u95a2\u6570\u3067\u3042\u308b. Hahn-Banach\u306e\u5b9a\u7406\u306f\u7dda\u5f62\u90e8\u5206\u7a7a\u9593\u4e0a\u306e\u7dda\u5f62\u6c4e\u95a2\u6570\u306f\u52a3\u7dda\u5f62\u6c4e\u95a2\u6570\u306b\u652f\u914d\u3055\u308c\u3066\u3044\u308b\u306e\u3067\u3042\u308c\u3070\u305d\u306e\u652f\u914d\u3092\u5d29\u3055\u305a\u306b\u5168\u7a7a\u9593\u306b\u62e1\u5f35\u3067\u304d\u308b\u3068\u3044\u3046\u4e3b\u5f35\u3067\u3042\u308b. \u6b63\u78ba\u306b\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3042\u308b. \n\n\\begin{thm}[Hahn-Banach\u306e\u62e1\u5f35\u5b9a\u7406] \\\\\n$V$\u3092$\\mathbb{R}$\u4e0a\u306e\u7dda\u5f62\u7a7a\u9593, $W \\subset V$\u3092\u7dda\u5f62\u90e8\u5206\u7a7a\u9593, $p : V \\rightarrow \\mathbb{R}$\u3092\u52a3\u7dda\u5f62\u6c4e\u95a2\u6570\u3068\u3059\u308b\u3068\u304d, \u7dda\u5f62\u6c4e\u95a2\u6570$f : W \\rightarrow \\mathbb{R}$\u304c$\\forall x \\in W, f(x) \\leq p(x)$ \u3092\u307f\u305f\u3059\u306a\u3089\u3070, \u7dda\u5f62\u6c4e\u95a2\u6570$F : V \\rightarrow \\mathbb{R}$\u3067\u3042\u3063\u3066, $W$\u4e0a\u3067$F = f$\u3068\u306a\u308a, $V$\u4e0a\u3067$F \\leq p$\u3068\u306a\u308b\u3082\u306e\u304c\u5b58\u5728\u3059\u308b. \n\\end{thm}\n\\begin{proof}\n  \u5bae\u5cf6\\cite{miyajima}\u3092\u53c2\u7167\u3055\u308c\u305f\u3044. \n\\end{proof}\n\u3053\u306e\u5b9a\u7406\u304b\u3089\u6b21\u306e\u5f62\u306eHahn-Banach\u306e\u62e1\u5f35\u5b9a\u7406\u304c\u5c0e\u3051\u308b. \n\\begin{thm}[Hahn-Banach\u306e\u62e1\u5f35\u5b9a\u74062] \\\\\n  $V$\u3092\u30ce\u30eb\u30e0\u7a7a\u9593\u3068\u3057, $W \\subset V$\u3092\u90e8\u5206\u7dda\u5f62\u7a7a\u9593\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $W$\u4e0a\u306e\u6709\u754c\u7dda\u5f62\u6c4e\u95a2\u6570$f:W \\rightarrow \\mathbb{R}$\u306b\u5bfe\u3057\u3066, $V$\u4e0a\u306e\u6709\u754c\u7dda\u5f62\u6c4e\u95a2\u6570$F$\u3067\n  \\[\n  F(x) = f(x) ~(x \\in W),~ \\lVert F \\rVert = \\lVert f \\rVert\n  \\]\n  \u3092\u6e80\u305f\u3059\u3082\u306e\u304c\u5b58\u5728\u3059\u308b. \n\\end{thm}\n\\begin{proof}\n  $x \\in V$\u306b\u5bfe\u3057\u3066$p(x) = \\lVert f \\rVert \\lVert x \\rVert$\u3068\u304a\u304f. \u660e\u3089\u304b\u306b$p$\u306f\u52a3\u7dda\u5f62\u6c4e\u95a2\u6570\u3067\u3042\u308a, $f(x) \\leq p(x) ~(x \\in W)$\u3067\u3042\u308b. \u5f93\u3063\u3066, Hahn-Banach\u306e\u62e1\u5f35\u5b9a\u7406\u3088\u308a$F(x) = f(x) ~(x \\in W)$\u304b\u3064$F(x) \\leq p(x) ~(x \\in V)$\u306a\u308b\u7dda\u5f62\u6c4e\u95a2\u6570$F:V \\rightarrow \\mathbb{R}$\u304c\u53d6\u308c\u308b. \u3053\u306e\u3068\u304d\n  \\[\n  -F(x) = F(-x) \\leq p(-x) = p(x) = \\lVert f \\rVert \\lVert x \\rVert ~~(x \\in V)\n  \\]\n  \u3067\u3042\u308b\u306e\u3067$F$\u306f\u6709\u754c\u3067$\\lVert F \\rVert \\leq \\lVert f \\rVert$\u3067\u3042\u308b. \u307e\u305f, \\[\n  |f(x)| = |F(x)| \\leq \\lVert F \\rVert \\lVert x \\rVert ~~~(x \\in W)\n  \\]\n  \u3067\u3042\u308b\u306e\u3067$\\lVert f \\rVert \\leq \\lVert F \\rVert$\u3067\u3042\u308b\u306e\u3067$\\lVert f \\rVert = \\lVert F \\rVert$\u3067\u3042\u308b. \n\\end{proof}\n\n\u672c\u8ad6\u6587\u3067\u7528\u3044\u308bHahn-Banach\u306e\u62e1\u5f35\u5b9a\u7406\u306e\u7cfb\u3092\u8ff0\u3079\u3088\u3046. \n\n\\begin{cor}\\label{HahnBanachCor}\n  $V$\u3092$\\mathbb{R}$\u4e0a\u306e\u30ce\u30eb\u30e0\u7a7a\u9593, $W \\subset V$\u3092\u7dda\u5f62\u90e8\u5206\u7a7a\u9593\u3068\u3059\u308b. \n  \u3053\u306e\u3068\u304d, \u3042\u308b$x_0 \\in V$\u304c$d := d(x_0,W) = \\inf_{x \\in W} \\lVert x - x_0 \\rVert > 0$\u3092\u6e80\u305f\u3059\u306a\u3089\u3070, \n  \\[\n  F(x) = 0 ~(x \\in W),~ F(x_0) = 1,~ \\lVert F \\rVert = \\frac{1}{d}\n  \\]\n  \u3092\u6e80\u305f\u3059\u6709\u754c\u7dda\u5f62\u6c4e\u95a2\u6570$F:V \\rightarrow \\mathbb{R}$\u304c\u5b58\u5728\u3059\u308b. \n\\end{cor}\n\\begin{proof}\n  $X := W + \\mathbb{R} x_0$\u3068\u304a\u304d, $f:X \\in w+ax_0 \\mapsto a \\in \\mathbb{R}$\u3068\u304a\u304f. $f$\u306fwell-defined\u3067\u3042\u308b. \u5b9f\u969b, $w_1+ax_0 = w_2 + bx_0$\u3068\u3059\u308b\u3068, $(b-a)x_0 = w_1-w_2 \\in W$\u3067\u3042\u308b\u306e\u3067$d(x_0,W)>0$\u3088\u308a$a=b$\u3067\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044.  \u660e\u3089\u304b\u306b$f$\u306f\u7dda\u5f62\u6c4e\u95a2\u6570\u3067\u3042\u308a$f(x) = 0 ~(x \\in W)$\u304b\u3064$f(x_0)=1$\u3092\u6e80\u305f\u3059. \u305d\u3057\u3066, $x \\in W, a \\mathbb{R} \\setminus \\{0\\}$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  \\lVert x + a x_0 \\rVert =|a| \\left\\lVert (-\\frac{1}{a}x) - x_0 \\right\\rVert \\leq |a| d\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b\u306e\u3067$|f(x+ax_0)| = |a| = \\lVert x+ax_0 \\rVert\/d$\u3067\u3042\u308b. \u3088\u3063\u3066, $f$\u306f\u6709\u754c\u3067\u3042\u308a$\\lVert f \\rVert \\leq 1\/d$\u3067\u3042\u308b. \u307e\u305f, $d=\\lim_{n\\rightarrow\\infty} \\lVert x_0 - x_n \\rVert$\u306a\u308b$W$\u306e\u5143\u306e\u5217$(x_n)$\u3092\u53d6\u308c\u3070, \n  \\[\n  1=f(-x_n + x_0) \\leq \\lVert f \\rVert \\lVert x_0 - x_n \\rVert \\rightarrow \\lVert f \\rVert d\n  \\]\n  \u3067\u3042\u308b\u306e\u3067$1\/d \\leq \\lVert f \\rVert$\u3067\u3042\u308b. \u3088\u3063\u3066$\\lVert f \\rVert = 1\/d$\u3067\u3042\u308b\u306e\u3067, Hahn-Banach\u306e\u62e1\u5f35\u5b9a\u7406$2$\u3088\u308a, \u6709\u754c\u7dda\u5f62\u6c4e\u95a2\u6570$F:V \\rightarrow \\mathbb{R}$\u3067\u3042\u3063\u3066\n  \\[\n  F(x) = f(x) ~(x \\in X),~ \\lVert F \\rVert = \\lVert f \\rVert = \\frac{1}{d}\n  \\]\n  \u306a\u308b\u3082\u306e\u304c\u5b58\u5728\u3059\u308b. \u3053\u306e$F$\u304c\u6c42\u3081\u308b\u3082\u306e\u3067\u3042\u308b. \n\\end{proof}\n\n\\begin{cor}\\label{HahnBanachCor1} \\\\\n  $V$\u3092$\\mathbb{R}$\u4e0a\u306e\u30ce\u30eb\u30e0\u7a7a\u9593, $W \\subset V$\u3092\u7dda\u5f62\u90e8\u5206\u7a7a\u9593\u3068\u3059\u308b. \u3082\u3057$W$\u306e$V$\u306b\u304a\u3051\u308b\u9589\u5305$\\overline{W}$\u304c$V$\u3068\u4e00\u81f4\u3057\u306a\u3051\u308c\u3070, $V$\u4e0a\u306e\u6709\u754c\u7dda\u5f62\u6c4e\u95a2\u6570$f \\neq 0$\u3067, $f = 0 ~ (\\mathrm{on} ~ \\overline{W})$\u306a\u308b\u3082\u306e\u304c\u5b58\u5728\u3059\u308b. \n\\end{cor}\n\\begin{proof}\n  $x_0 \\in V \\setminus \\overline{W}$\u3068\u3059\u308b\u3068$d(x_0,W) > 0$\u3067\u3042\u308b. \u5f93\u3063\u3066, \u3059\u3050\u4e0a\u306e\u7cfb\u304b\u3089$f(x_0)=1, f(x) = 0 ~(x\\in W)$\u306a\u308b\u7dda\u5f62\u6c4e\u95a2\u6570\n  $f:V \\rightarrow \\mathbb{R}$\u304c\u53d6\u308c\u308b. $x \\in \\overline{W}$\u306b\u5bfe\u3057\u3066$x$\u306b\u53ce\u675f\u3059\u308b\u5217$x_n \\in W$\u3092\u53d6\u308c\u3070, \n  \\[\n  |f(x)|=|f(x) - f(x_n)| = |f(x-x_n)| \\leq \\lVert f \\rVert \\lVert x-x_n \\rVert \\rightarrow 0\n  \\]\n  \u3067\u3042\u308b\u304b\u3089$f(x) = 0 ~(x \\in \\overline{W})$\u3067\u3042\u308b. \u3088\u3063\u3066\u3053\u306e$f$\u304c\u6c42\u3081\u308b\u3082\u306e\u3067\u3042\u308b. \n\\end{proof}\n\n\\begin{cor}\\label{HahnBanachCor2} \\\\\n  $x \\neq 0$\u306a\u3089\u3070$f(x) = \\lVert x \\rVert_X$\u304b\u3064$\\lVert f \\rVert = 1$\u306a\u308b\u6709\u754c\u7dda\u5f62\u6c4e\u95a2\u6570$f:X \\rightarrow \\mathbb{R}$\u304c\u5b58\u5728\u3059\u308b. \n\\end{cor}\n\\begin{proof}\n  $W = \\{0\\}$\u3068\u304a\u3051\u3070$x_0=x$\u3068\u3057\u3066\u7cfb\\ref{HahnBanachCor}\u306e\u4eee\u5b9a\u304c\u6e80\u305f\u3055\u308c\u308b. \u3086\u3048\u306b, $d = d(x,W) = \\lVert x \\rVert$\u3068\u304a\u304f\u3068, \u6709\u754c\u7dda\u5f62\u6c4e\u95a2\u6570$F:V \\rightarrow \\mathbb{R}$\u3067\n  \\[\n  F(x_0) = 1,~ \\lVert F \\rVert = \\frac{1}{d}\n  \\]\n  \u3092\u6e80\u305f\u3059\u3082\u306e\u304c\u53d6\u308c\u308b. \u3088\u3063\u3066, $f = \\lVert x \\rVert F$\u304c\u6c42\u3081\u308b\u3082\u306e\u3067\u3042\u308b. \n\\end{proof}\n\n\n\\subsection{Riesz-Markov-\u89d2\u8c37\u306e\u8868\u73fe\u5b9a\u7406}\n\u672c\u7bc0\u3067\u306f\u307e\u305a\u7b26\u53f7\u4ed8\u304d\u6e2c\u5ea6\u53ca\u3073\u305d\u308c\u306b\u3088\u308b\u7a4d\u5206\u306e\u5b9a\u7fa9\u3092\u975e\u8ca0\u5024\u306e\u6e2c\u5ea6\u306e\u77e5\u8b58\u3092\u524d\u63d0\u306b\u3057\u3066\u8ff0\u3079\u308b. \u305d\u306e\u5f8c, \u7b26\u53f7\u4ed8\u304dBorel\u6e2c\u5ea6\u306e\u5b9a\u7fa9\u3092\u8ff0\u3079, Riesz-Markov-\u89d2\u8c37\u306e\u8868\u73fe\u5b9a\u7406\u306e\u4e3b\u5f35\u3092\u8ff0\u3079\u308b. \u8a3c\u660e\u306b\u3064\u3044\u3066\u306f\u5bae\u5cf6\\cite{miyajima}\u306e\u7b2c7\u7ae0\u3092\u53c2\u7167\u3055\u308c\u305f\u3044. \n\n\\begin{defn}[\u7b26\u53f7\u4ed8\u304d\u6e2c\u5ea6] \\\\\n$(\\Omega, \\mathcal{F})$\u3092\u53ef\u6e2c\u7a7a\u9593\u3068\u3059\u308b. \u5199\u50cf$\\varphi : \\mathcal{F} \\rightarrow \\mathbb{R}$ \u304c\u6b21\u306e\u6761\u4ef6\u3092\u307f\u305f\u3059\u3068\u304d\u3092$\\varphi$\u3092$(\\Omega, \\mathcal{F})$\u4e0a\u306e\u7b26\u53f7\u4ed8\u304d\u6e2c\u5ea6\u3068\u547c\u3076:\n$\\mathcal{F}$\u306e\u5143\u306e\u5217$(A_n)$\u304c\u4e92\u3044\u306b\u7d20\u3067\u3042\u308b\u306a\u3089\u3070,  \n\\[\n\\varphi(\\bigcup_{n=1}^{\\infty}{A_n}) = \\sum_{n=1}^{\\infty}{\\varphi(A_n)}\n\\]\n\u3068\u306a\u308b. \n\\end{defn}\n\n\u4ee5\u4e0b\u306e\u4e8b\u5b9f\u306b\u57fa\u3065\u304d\u7b26\u53f7\u4ed8\u304d\u6e2c\u5ea6\u306b\u3088\u308b\u7a4d\u5206\u3092\u5b9a\u7fa9\u3059\u308b. \n\\begin{prop} \\\\\n$\\varphi$\u3092\u53ef\u6e2c\u7a7a\u9593$(\\Omega, \\mathcal{F})$\u4e0a\u306e\u7b26\u53f7\u4ed8\u304d\u6e2c\u5ea6\u3068\u3059\u308b\u3068\u304d, $A \\in \\mathcal{F}$\u306b\u5bfe\u3057\u3066, \n$$\n\\varphi^{+}(A) = \\sup \\\\{\\varphi(B) \\mid B \\in \\mathcal{F}, B \\subset A\\\\}, \\\\\n\\varphi^{-}(A) = -\\inf \\\\{\\varphi(B) \\mid B \\in \\mathcal{F}, B \\subset A\\\\}\n$$\u3068\u304a\u304f\u3068, $\\varphi^{+}, \\varphi^{-}$\u306f$(\\Omega, \\mathcal{F})$\u4e0a\u306e\u6709\u9650\u6e2c\u5ea6\u3068\u306a\u308a, $\\varphi = \\varphi^{+} - \\varphi^{-}$ \u304c\u6210\u7acb\u3059\u308b. \n\\end{prop}\n\n\\begin{defn} \\\\\n$\\varphi$\u3092\u53ef\u6e2c\u7a7a\u9593$(\\Omega, \\mathcal{F})$\u4e0a\u306e\u7b26\u53f7\u4ed8\u304d\u6e2c\u5ea6\u3068\u3059\u308b\u3068\u304d, $f \\in L^1(\\Omega, \\mathcal{F}, \\varphi^{+}) \\cap L^1(\\Omega, \\mathcal{F}, \\varphi^{-})$\u306b\u5bfe\u3057\u3066, $\\varphi$\u306b\u95a2\u3059\u308b$f$\u306e\u7a4d\u5206\u3092\u4ee5\u4e0b\u3067\u5b9a\u3081\u308b.  $$\\int_{\\Omega}{f}{d\\varphi} := \\int_{\\Omega}{f}{d\\varphi^{+}} - \\int_{\\Omega}{f}{d\\varphi^{-}}$$\n\\end{defn}\n\n\u6b21\u306b\u6b63\u5247\u7b26\u53f7\u4ed8\u304dBorel\u6e2c\u5ea6\u306e\u5b9a\u7fa9\u3092\u8ff0\u3079\u308b. \n\n\\begin{defn}[\u6b63\u5247\u7b26\u53f7\u4ed8\u304dBorel\u6e2c\u5ea6] \\\\\n\u4f4d\u76f8\u7a7a\u9593$(\\Omega, \\mathcal{F})$\u4e0a\u306e\u7b26\u53f7\u4ed8\u304d\u6e2c\u5ea6$\\varphi : \\mathcal{F} \\rightarrow \\mathbb{R}$ \u306f$\\varphi^{+}, \\varphi^{-}$\u304c\u3068\u3082\u306b$(\\Omega, \\mathcal{F})$\u4e0a\u306e\u6b63\u5247Borel\u6e2c\u5ea6\u306b\u306a\u308b\u3068\u304d, $(\\Omega, \\mathcal{F})$\u4e0a\u306e\u6b63\u5247\u7b26\u53f7\u4ed8\u304d\u6e2c\u5ea6\u3068\u547c\u3070\u308c\u308b. \n\\end{defn}\n\n\u306a\u304a, \u6587\u8108\u304b\u3089\u63a8\u5bdf\u53ef\u80fd\u306a\u5834\u5408\u306f$S$\u306b\u5165\u3063\u3066\u3044\u308b\u4f4d\u76f8\u3092\u660e\u793a\u305b\u305a\u5358\u306b$S$\u4e0a\u306e\u6b63\u5247\u7b26\u53f7\u4ed8\u304d\u6e2c\u5ea6\u3068\u547c\u3076\u3053\u3068\u3082\u3042\u308b. \u4ee5\u4e0a\u306e\u5b9a\u7fa9\u306e\u3082\u3068\u4ee5\u4e0b\u304c\u6210\u308a\u7acb\u3064. \n\n\\begin{thm}[Riesz-Markov-\u89d2\u8c37\u306e\u8868\u73fe\u5b9a\u7406] \\\\\n$S$\u3092\u30b3\u30f3\u30d1\u30af\u30c8Hausdorff\u7a7a\u9593\u3068\u3057, $C(S)$\u3092$S$\u4e0a\u306e\u5b9f\u6570\u5024\u9023\u7d9a\u95a2\u6570\u5168\u4f53\u306e\u96c6\u5408\u3068\u3059\u308b\u3068\u304d, \u4efb\u610f\u306e\u6709\u754c\u7dda\u5f62\u6c4e\u95a2\u6570$\\varphi : C(S) \\rightarrow \\mathbb{R}$\u306b\u5bfe\u3057\u3066, $S$\u4e0a\u306e\u6b63\u5247\u7b26\u53f7\u4ed8\u304dBorel\u6e2c\u5ea6$\\mu$\u3067\u3042\u3063\u3066, $$\\forall f \\in C(S), \\\\ \\varphi(f) = \\int_{S}{f}{d\\mu}$$\u3092\u307f\u305f\u3059\u3082\u306e\u304c\u4e00\u610f\u7684\u306b\u5b58\u5728\u3059\u308b. \n\\end{thm}\n\n\\subsection{Stone-Weierstrass\u306e\u5b9a\u7406}\n\u672c\u7bc0\u3067\u306fStone-Weierstrass\u306e\u5b9a\u7406\u3068\u3044\u3046, \u9023\u7d9a\u95a2\u6570\u7a7a\u9593\u306b\u304a\u3051\u308b\u4e00\u69d8\u53ce\u675f\u306e\u610f\u5473\u3067\u306e\u7a20\u5bc6\u6027\u306b\u95a2\u3059\u308b\u5b9a\u7406\u306e\u4e3b\u5f35\u3092\u8ff0\u3079\u308b. \u8a3c\u660e\u306f\u5bae\u5cf6\\cite{miyajima}\u307e\u305f\u306f\u65b0\u4e95\\cite{AraiFourier}\u306e\u4ed8\u9332\u3092\u53c2\u7167\u3055\u308c\u305f\u3044. \n\u306a\u304a, \u672c\u7bc0\u3067\u306f$S$\u3092\u30b3\u30f3\u30d1\u30af\u30c8Hausdorff\u7a7a\u9593\u3068\u3057, $C(S,\\mathbb{K})$\u3067$S$\u4e0a\u306e$\\mathbb{K}$-\u5024\u9023\u7d9a\u95a2\u6570\u5168\u4f53\u306e\u96c6\u5408\u3092\u8868\u3059($K = \\mathbb{R}, \\mathbb{C}$\u3068\u3059\u308b).\n$C(S,\\mathbb{K})$\u306f$\\|f\\| = \\sup \\{ f(x) \\mid x \\in S\\}$\u306b\u3088\u308a$\\mathbb{K}$\u4e0a\u306e\u30ce\u30eb\u30e0\u7a7a\u9593\u306b\u306a\u308b. \n\n\\begin{defn}[$C(S,\\mathbb{K})$\u306e\u90e8\u5206\u4ee3\u6570] \\\\\n$A \\subset C(S,\\mathbb{K})$\u304c\u90e8\u5206\u4ee3\u6570\u3067\u3042\u308b\u3068\u306f, $A$\u304c$C(S,\\mathbb{K})$\u306e$\\mathbb{K}$\u4e0a\u306e\u7dda\u5f62\u90e8\u5206\u7a7a\u9593\u3067\u3042\u308a, \u304b\u3064\u4efb\u610f\u306e$ f, g \\in A$\u306b\u3064\u3044\u3066\u7a4d$fg$\u304c$A$\u306e\u5143\u306b\u306a\u308b\u3053\u3068\u3092\u3044\u3046. \n\\end{defn}\n\n$C(S,\\mathbb{K})$\u306e\u90e8\u5206\u96c6\u5408$A$\u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d, $A$\u3092\u542b\u3080(\u5305\u542b\u95a2\u4fc2\u306e\u610f\u5473\u3067)\u6700\u5c0f\u306e\u90e8\u5206\u4ee3\u6570\u304c\u5b58\u5728\u3059\u308b($A$\u3092\u542b\u3080\u90e8\u5206\u4ee3\u6570\u5168\u4f53\u306e\u5171\u901a\u90e8\u5206\u3092\u53d6\u308c\u3070\u3088\u3044). \u305d\u308c\u3092$A$\u3067\u751f\u6210\u3055\u308c\u305f\u90e8\u5206\u4ee3\u6570\u3068\u547c\u3076.  Stone-Weierstrass\u306e\u5b9a\u7406\u306f\u751f\u6210\u3055\u308c\u305f\u90e8\u5206\u4ee3\u6570\u306e\u7a20\u5bc6\u6027\u306b\u3064\u3044\u3066\u8ff0\u3079\u305f\u3082\u306e\u3067\u3042\u308b. \n\n\\begin{thm}[Stone-Weierstrass\u306e\u5b9a\u7406(\u5b9f\u6570\u5f62)] \\\\\n$A \\subset C(S,\\mathbb{R})$\u304c\u6b21\u306e2\u3064\u306e\u6761\u4ef6\u3092\u307f\u305f\u3057\u3066\u3044\u308b\u3068\u3059\u308b. \n\\[\n\\begin{aligned}\n&(1)\\mbox{\u6052\u7b49\u7684\u306b1\u3092\u53d6\u308b\u95a2\u6570} \\mathbf{1} \\mbox{\u306f} A \\mbox{\u306b\u5c5e\u3059\u308b}.  \\\\\n&(2)\\mbox{\u4efb\u610f\u306e\u76f8\u7570\u306a\u308b} x, y \\in S \\mbox{\u306b\u5bfe\u3057\u3066, } f(x) \\neq f(y) \\mbox{\u306a\u308b} f \\in A \\mbox{\u304c\u5b58\u5728\u3059\u308b}. \n\\end{aligned}\n\\]\n\u3053\u306e\u3068\u304d$A$\u304c\u751f\u6210\u3059\u308b\u90e8\u5206\u4ee3\u6570\u306e\u9589\u5305\u306f$C(S,\\mathbb{R})$\u3068\u4e00\u81f4\u3059\u308b. \u3059\u306a\u308f\u3061, \u4efb\u610f\u306e$f \\in C(S,\\mathbb{R})$\u3068$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, $A$\u304c\u751f\u6210\u3059\u308b\u90e8\u5206\u4ee3\u6570\u306e\u5143$g$\u3067, $|f-g| < \\varepsilon$\u3092\u307f\u305f\u3059\u3082\u306e\u304c\u53d6\u308c\u308b. \n\\end{thm}\n\\begin{cor}\n  $K \\subset \\mathbb{R}$\u3092\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u3068\u3059\u308b\u3068\u304d, 1\u5909\u6570\u306e$\\mathbb{R}$\u4fc2\u6570\u591a\u9805\u5f0f\u95a2\u6570\u3092$K$\u4e0a\u306b\u5236\u9650\u3057\u305f\u3082\u306e\u5168\u4f53\u306e\u96c6\u5408\u3092$A$\u3068\u304a\u304f\u3068, \u660e\u3089\u304b\u306b$A$\u306f\u4e0a\u306e2\u6761\u4ef6\u3092\u307f\u305f\u3059\u306e\u3067Stone-Weierstrass\u306e\u5b9a\u7406\u304b\u3089$A$\u304c\u751f\u6210\u3059\u308b\u90e8\u5206\u4ee3\u6570\u306e\u9589\u5305\u306f$C(K,\\mathbb{R})$\u3068\u4e00\u81f4\u3059\u308b. \u3068\u3053\u308d\u304c$A$\u306f\u90e8\u5206\u4ee3\u6570\u306a\u306e\u3067$A$\u304c\u751f\u6210\u3059\u308b\u90e8\u5206\u4ee3\u6570\u306f$A$\u306b\u306a\u308b. \u3088\u3063\u3066, $K$\u4e0a\u306e\u5b9f\u6570\u5024\u9023\u7d9a\u95a2\u6570\u306f\u591a\u9805\u5f0f\u306b\u3088\u308a\u4e00\u69d8\u8fd1\u4f3c\u3067\u304d\u308b. \n\\end{cor}\n\\begin{thm}[Stone-Weierstrass\u306e\u5b9a\u7406(\u8907\u7d20\u6570\u5f62)] \\\\\n$A \\subset C(S,\\mathbb{C})$\u304c\u6b21\u306e2\u3064\u306e\u6761\u4ef6\u3092\u307f\u305f\u3057\u3066\u3044\u308b\u3068\u3059\u308b. \n\\[\n\\begin{aligned}\n&(1)\\mbox{\u6052\u7b49\u7684\u306b1\u3092\u53d6\u308b\u95a2\u6570} \\mathbf{1} \\mbox{\u306f} A \\mbox{\u306b\u5c5e\u3059\u308b}.  \\\\\n&(2)\\mbox{\u4efb\u610f\u306e\u76f8\u7570\u306a\u308b} x, y \\in S \\mbox{\u306b\u5bfe\u3057\u3066, } f(x) \\neq f(y) \\mbox{\u306a\u308b} f \\in A \\mbox{\u304c\u5b58\u5728\u3059\u308b}. \\\\\n&(3)f \\in A \\Rightarrow \\overline{f} \\in A\n\\end{aligned}\n\\]\n\u3053\u306e\u3068\u304d$A$\u304c\u751f\u6210\u3059\u308b\u90e8\u5206\u4ee3\u6570\u306e\u9589\u5305\u306f$C(S,\\mathbb{C})$\u3068\u4e00\u81f4\u3059\u308b. \u3059\u306a\u308f\u3061, \u4efb\u610f\u306e$f \\in C(S,\\mathbb{C})$\u3068$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, $A$\u304c\u751f\u6210\u3059\u308b\u90e8\u5206\u4ee3\u6570\u306e\u5143$g$\u3067, $|f-g| < \\varepsilon$\u3092\u307f\u305f\u3059\u3082\u306e\u304c\u53d6\u308c\u308b. \n\\end{thm}\n\\begin{cor}\n  $K \\subset \\mathbb{R}^n$\u3092\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u3068\u3057, $m \\in \\mathbb{Z}^n$\u306b\u5bfe\u3057\u3066$e_{m}:K \\ni x \\mapsto \\exp(i\\ip<{m,x}>) \\in \\mathbb{C}$\u3068\u304a\u304f. \u3053\u306e\u3068\u304d$A := \\mathrm{span}\\{e_m\\}_{m \\in \\mathbb{Z}^n}$\u3068\u304a\u304f\u3068, \u660e\u3089\u304b\u306b$A$\u306f\u4e0a\u306e3\u6761\u4ef6\u3092\u307f\u305f\u3059\u306e\u3067Stone-Weierstrass\u306e\u5b9a\u7406\u304b\u3089$A$\u304c\u751f\u6210\u3059\u308b\u90e8\u5206\u4ee3\u6570\u306e\u9589\u5305\u306f$C(S)$\u3068\u4e00\u81f4\u3059\u308b.  \u3068\u3053\u308d\u304c$A$\u306f\u90e8\u5206\u4ee3\u6570\u306a\u306e\u3067$A$\u304c\u751f\u6210\u3059\u308b\u90e8\u5206\u4ee3\u6570\u306f$A$\u306b\u306a\u308b. \u3088\u3063\u3066,  $K$\u4e0a\u306e\u8907\u7d20\u6570\u5024\u9023\u7d9a\u95a2\u6570\u306f$A$\u306e\u5143\u306b\u3088\u308a\u4e00\u69d8\u8fd1\u4f3c\u3067\u304d\u308b. \n\\end{cor}\n\nRiesz-Markov-\u89d2\u8c37\u306e\u5b9a\u7406\u3068Stone-Weierstrass\u306e\u5b9a\u7406\u304b\u3089\u6b21\u306e\u3053\u3068\u304c\u308f\u304b\u308b. \n\n\\begin{cor}\\label{CorforChuiandLi}\n$K \\subset \\mathbb{R}^n$\u3092\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u3068\u3057,  $\\mu$\u3092\u90e8\u5206\u7a7a\u9593$K$\u4e0a\u306e\u6b63\u5247\u7b26\u53f7\u4ed8\u304dBorel\u6e2c\u5ea6\u3068\u3059\u308b\u3068, \n\\[\n\\left(\\forall m \\in \\mathbb{Z}^n, ~\\int_{K}{\\exp(i \\left<x, m\\right>)}{d\\mu(x)} = 0 \\right) \\Rightarrow \\mu = 0\n\\]\n\u304c\u6210\u308a\u7acb\u3064. \n\\end{cor}\n\\begin{proof}\n  \u4efb\u610f\u306b$f \\in C(K)$\u3068$\\varepsilon$\u3092\u3068\u308b. \u3059\u3050\u4e0a\u306e\u7cfb\u3088\u308a$c_k \\in \\mathbb{C}, m_k \\in \\mathbb{Z}^n$ $(k=1,\\ldots,N)$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  \\forall x \\in K, ~ \\left|f(x) - \\sum_{k=1}^N c_k e_{m_k}(x) \\right| < \\varepsilon\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, \n  \\[\n  \\begin{aligned}\n  |\\int_{K} f(x) d\\mu(x)| \n  &= |\\int_{K} f(x) - \\sum_{k=1}^N c_k e_{m_k}(x) d\\mu(x)| \\\\\n  &\\leq \\int_K \\left|f(x) - \\sum_{k=1}^N c_k e_{m_k}(x) \\right| d|\\mu|(x)\n  \\leq |\\mu|(K) \\varepsilon\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. $\\varepsilon > 0$\u306f\u4efb\u610f\u306a\u306e\u3067$\\int_{K} f(x) d\\mu(x) = 0$\u3067\u3042\u308b. \u3068\u3053\u308d\u304c, $K$\u4e0a\u306e\u6b63\u5247\u7b26\u53f7\u4ed8\u304dBorel\u6e2c\u5ea6$\\nu = 0$\u3082\u3053\u308c\u3092\u6e80\u305f\u3059. \u3088\u3063\u3066, Riesz-Markov-\u89d2\u8c37\u306e\u5b9a\u7406\u306e\u4e00\u610f\u6027\u306e\u90e8\u5206\u306b\u3088\u308a$\\mu = \\nu = 0$\u3067\u3042\u308b. \n\\end{proof}\n\n\\subsection{\u7573\u307f\u8fbc\u307f\u7a4d\u306e\u6027\u8cea}\n\u672c\u7bc0\u3067\u306f, $C_0^{\\infty}(\\mathbb{R})$\u3068$L_{\\mathrm{loc}}^1(\\mathbb{R})$\u306e\u7573\u307f\u8fbc\u307f\u304c$C^{\\infty}(\\mathbb{R})$\u306b\u306a\u308b\u3053\u3068, $L^1(\\mathbb{R}^N)$\u3068$L^p(\\mathbb{R}^N)$\u306e\u7573\u307f\u8fbc\u307f\u304c$L^p(\\mathbb{R}^N)$\u306b\u306a\u308b\u3053\u3068\u3092\u793a\u3059. \n\n\\begin{prop}\\label{L1andLpConvolution}\n  $1 \\leq p < \\infty$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $\\varphi \\in L^1(\\mathbb{R}^n)$\u3068\u3057$f \\in L^p(\\mathbb{R}^n)$\u3068\u3059\u308b\u3068$\\varphi * f \\in L^p(\\mathbb{R}^n)$\u3067\u3042\u308a, \n  \\[\n   \\lVert \\varphi * f \\rVert_{L^p} \\leq \\lVert \\varphi \\rVert_{L^1} \\lVert f \\rVert_{L^p}.\n  \\]\n\\end{prop}\n\\begin{proof}\n   $p=1$\u306e\u5834\u5408\u3092\u307e\u305a\u8003\u3048\u308b. \u3053\u306e\u3068\u304d, \n   \\[\n   \\begin{aligned}\n   \\int_{\\mathbb{R}^n} |(\\varphi*f)(x)| dx\n   &= \\int_{\\mathbb{R}^n} \\left| \\int_{\\mathbb{R}^n} \\varphi(x-y)f(y) dy \\right| dx \\\\\n   &= \\int_{\\mathbb{R}^n} \\int_{\\mathbb{R}^n} |\\varphi(x-y)| |f(y)| dy dx \\\\\n   &= \\int_{\\mathbb{R}^n} \\int_{\\mathbb{R}^n} |\\varphi(x-y)| |f(y)| dx dy ~~~~(\\mbox{Fubini\u306e\u5b9a\u7406}) \\\\\n   &= \\int_{\\mathbb{R}^n} \\lVert \\varphi \\rVert_{L^1} |f(y) dy \\\\\n   &= \\lVert \\varphi \\rVert_{L^1} \\lVert f \\rVert_{L^1}\n   \\end{aligned}\n   \\]\n   \u3067\u3042\u308b\u306e\u3067\u6210\u7acb\u3059\u308b. \u6b21\u306b$p > 1$\u3068\u3057$1\/p + 1\/q = 1$\u3068\u3059\u308b. \u3059\u308b\u3068, \u4efb\u610f\u306e$x \\in \\mathbb{R}^n$\u306b\u5bfe\u3057\u3066, H\\\"{o}lder\u306e\u4e0d\u7b49\u5f0f(\u547d\u984c\\ref{HolderInequality})\u3088\u308a\n   \\[\n   \\begin{aligned}\n   |(\\varphi*f)(x)|\n   &= \\left| \\int_{\\mathbb{R}^n} \\varphi(x-y)f(y) dy \\right| \\\\\n   &\\leq  \\int_{\\mathbb{R}^n} |\\varphi(x-y)|^{1-1\/p}|\\varphi(x-y)|^{1\/p}|f(y)| dy \\\\\n   &\\leq \\left( \\int_{\\mathbb{R}^n} |\\varphi(x-y)| d\\mu(y)  \\right)^{1\/q} \\left( \\int_{\\mathbb{R}^n} |\\varphi(x-y)||f(y)|^p dy  \\right)^{1\/p} \\\\\n   &= \\lVert \\varphi \\rVert_{L^1}^{1\/q} \\left( \\int_{\\mathbb{R}^n} |\\varphi(x-y) | |f(x-y)|^p dy \\right)^{1\/p}\n   \\end{aligned}\n   \\]\n   \u3067\u3042\u308b\u304b\u3089, \n   \\[\n   \\begin{aligned}\n   &\\int_X \\left| \\int_{\\mathbb{R}^n}  \\varphi(x-y) f(y) dx \\right|^p dx\\\\\n   &\\leq \\int_{\\mathbb{R}^n} \\lVert \\varphi \\rVert_{L^1}^{p\/q} \\int_{\\mathbb{R}^n} |\\varphi(x-y)| |f(x-y)|^p dy dx \\\\\n   &= \\lVert \\varphi \\rVert_{L^1}^{p\/q} \\int_{\\mathbb{R}^n} \\int_{\\mathbb{R}^n} |\\varphi(x-y)||f(y)| dxdy ~~~~(\\because ~\\mbox{Fubini\u306e\u5b9a\u7406}) \\\\\n   &= \\lVert \\varphi \\rVert_{L^1}^{p\/q} \\int_{\\mathbb{R}^n} |f(y)|^p \\lVert \\varphi \\rVert_{L^1} dy \\\\\n   &= \\lVert \\varphi \\rVert_{L^1}^p \\lVert f \\rVert_{L^p}^p < \\infty ~~~~(\\because ~ 1+p\/q = p)\n   \\end{aligned}\n   \\]\n   \u3068\u306a\u308b. \u3088\u3063\u3066, $\\varphi * f \\in L^p(\\mathbb{R}^n)$\u3067\u3042\u308a, \n   \\[\n   \\lVert \\varphi * f \\rVert_{L^p} \\leq \\lVert \\varphi \\rVert_{L^1} \\lVert f \\rVert_{L^p}.\n   \\]\n\\end{proof}\n\\begin{prop}\\label{L1locandC0Convolution}\n  $\\varphi \\in C_0^{\\infty}(\\mathbb{R})$\u3068\u3057$f \\in L_{\\mathrm{loc}}^1(\\mathbb{R})$\u3068\u3059\u308b\u3068$\\varphi * f \\in C^{\\infty}(\\mathbb{R})$\u3067\u3042\u308a, \u4efb\u610f\u306e\u6574\u6570$k \\geq 0$\u306b\u5bfe\u3057\u3066$(\\varphi * f)^{(k)} = \\varphi^{(k)} * f$\u3068\u306a\u308b. \n\\end{prop}\n\\begin{proof}\n  $K = \\mathrm{supp}\\varphi$\u3068\u304a\u304f. \u4efb\u610f\u306e\u6574\u6570$k \\geq 0$\u306b\u5bfe\u3057\u3066$\\mathrm{supp}(\\varphi^{(k)}) \\subset K$\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b. \u547d\u984c\u3092$k$\u306b\u3064\u3044\u3066\u306e\u5e30\u7d0d\u6cd5\u3067\u793a\u305d\u3046. $k=0$\u3067\u306f\u81ea\u660e. $k$\u3067\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b. \u4efb\u610f\u306b$x_0 \\in \\mathbb{R}$\u3092\u53d6\u308a$a<b$\u3092$x_0 \\in [a,b]$\u3068\u306a\u308b\u3088\u3046\u306b\u53d6\u308b. $K' = [a,b] - K = \\{ x-y \\mid x \\in [a,b],y \\in K \\}$\u3068\u304a\u304f\u3068$K'$\u306f\u30b3\u30f3\u30d1\u30af\u30c8\u3067\u3042\u308a, $x \\in (a,b)$\u306b\u5bfe\u3057\u3066\n  \\[\n  (\\varphi^{(k)} * f)(x) = \\int_{\\mathbb{R}} \\varphi^{(k)}(x-y) f(y) dy = \\int_{K'} \\varphi^{(k)}(x-y) f(y) dy\n  \\]\n  \u3068\u306a\u308b. \u3044\u307e, $g(x,y) = \\varphi^{(k)}(x-y)f(y) ~~(x \\in (a,b), y \\in K')$\u3068\u304a\u3051\u3070, \n  \\[\n  \\left| \\frac{\\partial g}{\\partial x}(x,y) \\right| = M |f(y)| ~~~~(x \\in (a,b), y \\in K')\n  \\]\n  \u3068\u306a\u308b. \u3053\u3053\u3067, $M = \\sup_{t \\in \\mathbb{R}} |\\varphi^{(k+1)}(t)| < \\infty $\u3067\u3042\u308b. $f$\u306f$K'$\u4e0a\u53ef\u7a4d\u5206\u3067\u3042\u308b\u306e\u3067\u7a4d\u5206\u8a18\u53f7\u4e0b\u306e\u5fae\u5206(\u4ed8\u9332\u306e\u5b9a\u7406\\ref{DiffinIntegral})\u3088\u308a$(a,b)$\u4e0a\u3067\n  \\[\n  F(x) := (\\varphi * f)^{(k)}(x) = (\\varphi^{(k)} * f)(x) = \\int_{K'} \\varphi^{(k)}(x-y) f(y) dy = \\int_{K'} g(x,y) dy\n  \\]\n  \u306f\u5fae\u5206\u53ef\u80fd\u3067, \n  \\[\n  F'(x) = \\int_{K'} \\frac{\\partial g}{\\partial x}(x,y) dy = \\int_{K'} \\varphi^{(k+1)}(x-y) f(y) dy = (\\varphi^{(k+1)} * f)(x)\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, $(\\varphi * f)^{(k+1)} = \\varphi^{(k+1)} * f$\u3067\u3042\u308b. \u3053\u308c\u3067\u793a\u305b\u305f. \n\\end{proof}\n\n\\subsection{$\\mathbb{R}^n$\u4e0a\u306e\u6709\u9650Borel\u6e2c\u5ea6\u306e\u6b63\u5247\u6027}\n\u672c\u7bc0\u3067\u306f$(\\mathbb{R}^n,\\mathcal{B}_{\\mathbb{R}^n})$\u4e0a\u306e\u6709\u9650\u6e2c\u5ea6\u304c\u6b21\u306e\u610f\u5473\u3067\u6b63\u5247\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059. \n\\begin{defn}[\u6b63\u5247Borel\u6e2c\u5ea6] \\\\\n\u4f4d\u76f8\u7a7a\u9593$(X, \\mathcal{O})$\u4e0a\u306eBorel\u6e2c\u5ea6$\\mu$\u306f\u4ee5\u4e0b\u306e\u6761\u4ef6\u3092\u307f\u305f\u3059\u3068\u304d\u6b63\u5247\u3067\u3042\u308b\u3068\u8a00\u308f\u308c\u308b:\n$$\\forall A \\in \\mathcal{B}_X, ~~ \\mu(A) = \\inf \\{\\mu(U) \\mid U \\supset A, \\mathrm{open}\\} = \\sup \\{\\mu(K) \\mid K \\subset A, \\mathrm{cpt. }\\}. $$\n\\end{defn}\n\u305f\u3060\u3057, \u4f4d\u76f8\u7a7a\u9593\u4e0a\u306eBorel $\\sigma$-\u52a0\u6cd5\u65cf, \u304a\u3088\u3073Borel\u6e2c\u5ea6\u306e\u5b9a\u7fa9\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3042\u308b. \n\\begin{defn}[\u751f\u6210\u3055\u308c\u308b$\\sigma$-\u52a0\u6cd5\u65cf] \\\\\n  $X$\u3092\u96c6\u5408\u3068\u3057, $\\mathcal{A}_0 \\subset \\mathcal{P}(X)$\u3068\u3059\u308b($\\mathcal{P}(X)$\u306f$X$\u306e\u90e8\u5206\u96c6\u5408\u5168\u4f53\u306e\u96c6\u5408). \u3053\u306e\u3068\u304d, $\\mathcal{A}_0$\u3067\u751f\u6210\u3055\u308c\u308b$\\sigma$-\u52a0\u6cd5\u65cf$\\sigma[\\mathcal{A}_0]$\u3092\n  \\[\n  \\sigma[\\mathcal{A}_0] := \\bigcap \\{ \\mathcal{A} \\mid \\mathcal{A} \\supset \\mathcal{A}_0, \\mathcal{A}\\mbox{\u306f}X\\mbox{\u4e0a\u306e}\\sigma\\mbox{-\u52a0\u6cd5\u65cf} \\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u660e\u3089\u304b\u306b$\\sigma[\\mathcal{A}_0]$\u306f$X$\u4e0a\u306e$\\sigma$-\u52a0\u6cd5\u65cf\u3067\u3042\u308b. \n\\end{defn}\n\\begin{defn}[Borel $\\sigma$-\u52a0\u6cd5\u65cf, Borel\u6e2c\u5ea6] \\\\\n  $(X,\\mathcal{O})$\u3092\u4f4d\u76f8\u7a7a\u9593\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $X$\u4e0a\u306eBorel $\\sigma$-\u52a0\u6cd5\u65cf$\\mathcal{B}_X$\u3092$\\mathcal{B}_X = \\sigma[\\mathcal{O}]$\u3068\u5b9a\u7fa9\u3059\u308b. \u305d\u3057\u3066, $X$\u4e0a\u306eBorel\u6e2c\u5ea6\u3068\u306f, \u53ef\u6e2c\u7a7a\u9593$(X,\\mathcal{B}_X)$\u4e0a\u306e\u6e2c\u5ea6\u306e\u3053\u3068\u3092\u3044\u3046. \n\\end{defn}\n\\begin{rem}\n  Borel $\\sigma$-\u52a0\u6cd5\u65cf\u306f\u9589\u96c6\u5408\u5168\u4f53\u306e\u96c6\u5408$\\mathcal{D}$\u304b\u3089\u751f\u6210\u3055\u308c\u308b. \u5b9f\u969b, $\\mathcal{O} \\subset \\mathcal{A}$\u306a\u308b$\\sigma$-\u52a0\u6cd5\u65cf$\\mathcal{A}$\u306f$E \\in \\mathcal{A} \\Rightarrow X \\setminus E \\in \\mathcal{A}$\u3092\u307f\u305f\u3059\u306e\u3067$\\mathcal{D} \\subset \\mathcal{A}$\u3092\u307f\u305f\u3057, $\\mathcal{D} \\subset \\mathcal{A}$\u306a\u308b$\\sigma$-\u52a0\u6cd5\u65cf$\\mathcal{A}$\u306f$E \\in \\mathcal{A} \\Rightarrow X \\setminus E \\in \\mathcal{A}$\u3092\u307f\u305f\u3059\u306e\u3067$\\mathcal{O} \\subset \\mathcal{A}$\u3092\u307f\u305f\u3059. \n\\end{rem}\n\\begin{lem}\\label{PropOfDist}\n  $(X,d)$\u3092\u8ddd\u96e2\u7a7a\u9593\u3068\u3057, $A \\subset X$\u306f\u7a7a\u3067\u306a\u3044\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $x \\in X$\u306b\u5bfe\u3057\u3066\n  \\[\n  \\mathrm{dist}(x,A) := \\inf \\{d(x,y) \\mid y \\in A\\}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b\u3068, $X \\ni x \\mapsto \\mathrm{dist}(x,A) \\in \\mathbb{R}$\u306f\u9023\u7d9a\u3067\u3042\u308b. \u307e\u305f, $x \\in X$\u306b\u5bfe\u3057\u3066$\\mathrm{dist}(x,A) = 0$\u3068$x \\in \\mathrm{cl}(A)$\u306f\u540c\u5024\u3067\u3042\u308b. \u7279\u306b,  $A$\u304c\u9589\u96c6\u5408\u3067\u3042\u308b\u3068\u304d, $\\mathrm{dist}(x,A) = 0$\u3068$x \\in A$\u306f\u540c\u5024\u3067\u3042\u308b.\n\\end{lem}\n\\begin{proof}\n  \u4efb\u610f\u306e$x_1,x_2 \\in X, y \\in A$\u306b\u5bfe\u3057\u3066\n  \\[\n  \\mathrm{dist}(x_1,A) \\leq d(x_1,y) \\leq d(x_1,x_2)+d(x_2,y)\n  \\]\n  \u3067\u3042\u308b\u306e\u3067, $\\mathrm{dist}(x_1,A) - d(x_1,x_2) \\leq d(x_2,y)$\u3068\u306a\u308b. \u305d\u3053\u3067, \u5de6\u8fba\u304c$y$\u306b\u4f9d\u5b58\u3057\u3066\u3044\u306a\u3044\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066$y$\u306b\u95a2\u3057\u3066\u4e0b\u9650\u3092\u53d6\u308b\u3068\n  \\[\n  \\mathrm{dist}(x_1,A) - d(x_1,x_2) \\leq \\mathrm{dist}(x_2,A)\n  \\]\n  \u304c\u5f97\u3089\u308c\u308b. \u3088\u3063\u3066, $\\mathrm{dist}(x_1,A)- \\mathrm{dist}(x_2,A) \\leq d(x_1,x_2)$\u3068\u306a\u308b. $x_1$\u3068$x_2$\u306e\u5f79\u5272\u3092\u4ea4\u63db\u3059\u308b\u3068, $\\mathrm{dist}(x_2,A)-\\mathrm{dist}(x_1,A) \\leq d(x_2,x_1)$\u304c\u5f97\u3089\u308c\u308b. \u4ee5\u4e0a\u304b\u3089, \n  \\[\n  |\\mathrm{dist}(x_1,A)-\\mathrm{dist}(x_2,A)| \\leq d(x_1,x_2)\n  \\]\n  \u3068\u306a\u308a, \u3053\u308c\u3088\u308a\u9023\u7d9a\u6027\u304c\u5f93\u3046. \u6b21\u306b$x \\in X$\u306b\u5bfe\u3057\u3066$\\mathrm{dist}(x,A) = 0$\u3068\u3059\u308b. \u3059\u308b\u3068, \u4efb\u610f\u306e$n$\u306b\u5bfe\u3057\u3066, $d(x,y_n) < 1\/n$\u306a\u308b$y_n \\in A$\u304c\u53d6\u308c\u308b. \u3053\u306e\u3068\u304d, $y_n \\rightarrow x$\u3067\u3042\u308b\u306e\u3067$x \\in \\mathrm{cl}(A)$\u3068\u306a\u308b. $x \\in \\mathrm{cl}(A)$\u306e\u3068\u304d$d(x,A) =  0$\u3068\u306a\u308b\u3053\u3068\u306b\u3064\u3044\u3066\u306f\u4e0a\u306e\u8b70\u8ad6\u3092\u9006\u5411\u304d\u306b\u3059\u308c\u3070\u3088\u3044. \n\\end{proof}\n\\begin{prop}\n  \u8ddd\u96e2\u7a7a\u9593$X$\u4e0a\u306e\u6709\u9650Borel\u6e2c\u5ea6$\\mu$\u306f\u6b21\u306e\u610f\u5473\u3067\u6b63\u5247\u3067\u3042\u308b: \u4efb\u610f\u306e$A \\in \\mathcal{B}_X$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  \\mu(A) \n  &= \\sup\\{\\mu(C) \\mid C \\subset A, C\\mbox{\u306f\u9589\u96c6\u5408}\\} \\\\\n  &= \\inf\\{\\mu(U) \\mid U \\supset A, U\\mbox{\u306f\u958b\u96c6\u5408} \\}\n  \\end{aligned}\n  \\]\n  \u304c\u6210\u7acb\u3059\u308b. \n\\end{prop}\n\\begin{proof}\n  $A \\in \\mathcal{B}_X$\u304b\u3064\n  \\[\n  \\begin{aligned}\n  \\mu(A) \n  &= \\sup\\{\\mu(C) \\mid C \\subset A, C\\mbox{\u306f\u9589\u96c6\u5408}\\} \\\\\n  &= \\inf\\{\\mu(U) \\mid U \\supset A, U\\mbox{\u306f\u958b\u96c6\u5408} \\}\n  \\end{aligned}\n  \\]\n  \u306a\u308b\u96c6\u5408$A$\u5168\u4f53\u306e\u96c6\u5408\u3092$\\mathcal{R}$\u3068\u304a\u304f. \u3059\u308b\u3068, $\\mathcal{R}$\u306f$X$\u4e0a\u306e$\\sigma$-\u52a0\u6cd5\u65cf\u3067\u3042\u308b. \u307e\u305a, $\\emptyset \\in \\mathcal{R}$\u306f\u660e\u3089\u304b. \u6b21\u306b, $A \\in \\mathcal{R}$\u3068\u3059\u308b\u3068$X \\setminus A \\in \\mathcal{R}$\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059. \u4efb\u610f\u306b$\\varepsilon > 0$\u3092\u3068\u308b. \u9589\u96c6\u5408$C \\subset A$\u3068\u958b\u96c6\u5408$U \\supset A$\u3092\n  \\[\n  \\mu(A) < \\mu(C) + \\varepsilon, \\mu(A) > \\mu(U) - \\varepsilon\n  \\]\n  \u3092\u307f\u305f\u3059\u3088\u3046\u306b\u53d6\u308b. \u3059\u308b\u3068, $X \\setminus C \\supset X \\setminus A$\u306f\u958b\u96c6\u5408\u3067\u3042\u308b\u306e\u3067, \n  \\[\n  \\begin{aligned}\n  \\mu(X \\setminus A) \n  &= \\mu(X) - \\mu(A) \\\\\n  &> \\mu(X) - \\mu(C) - \\varepsilon \\\\\n  &= \\mu(X \\setminus C) - \\varepsilon \\\\\n  &\\geq \\inf\\{\\mu(E) \\mid E \\supset X \\setminus A, E\\mbox{\u306f\u958b\u96c6\u5408}\\} - \\varepsilon\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308a, $X \\setminus U \\subset X \\setminus A$\u306f\u9589\u96c6\u5408\u306a\u306e\u3067, \n  \\[\n  \\begin{aligned}\n  \\mu(X \\setminus A)\n  &= \\mu(X) - \\mu(A) \\\\\n  &< \\mu(X) - \\mu(U) + \\varepsilon \\\\\n  &= \\mu(X \\setminus U) + \\varepsilon \\\\\n  &\\leq \\sup\\{\\mu(E) \\mid E \\subset X \\setminus A, E\\mbox{\u306f\u9589\u96c6\u5408}\\} + \\varepsilon\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, $\\varepsilon > 0$\u306e\u4efb\u610f\u6027\u306b\u3088\u3063\u3066$X \\setminus A \\in \\mathcal{R}$\u3068\u306a\u308b. \u6b21\u306b$\\mathcal{R}$\u306e\u5143\u306e\u5217$(A_n)_{n=1}^{\\infty}$\u306b\u5bfe\u3057\u3066$\\bigcup_{n=1}^{\\infty} A_n \\in \\mathcal{R}$\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3059. \u305d\u306e\u305f\u3081\u306b$\\varepsilon > 0$\u3092\u4efb\u610f\u306b\u3068\u308a, \u5404$n$\u306b\u5bfe\u3057\u3066, \u958b\u96c6\u5408$U_n$\u3068\u9589\u96c6\u5408$C_n$\u3092$C_n \\subset A_n \\subset U_n$\u304b\u3064\n  \\[\n  \\mu(U_n) - \\mu(A_n) < \\frac{\\varepsilon}{2^n},~~ \\mu(A_n) - \\mu(C_n) < \\frac{\\varepsilon}{2^n}\n  \\]\n  \u3068\u306a\u308b\u3088\u3046\u306b\u3068\u308b. \u3059\u308b\u3068, $U := \\bigcup_{n=1}^{\\infty} U_n \\supset \\bigcup_{n=1}^{\\infty} A_n =: A$\u306f\u958b\u96c6\u5408\u3067\u3042\u308a, \n  \\[\n  \\begin{aligned}\n  \\mu(U) - \\mu(A) \n  &= \\mu\\left((\\bigcup U_n) \\setminus (\\bigcup A_n)\\right) \\\\\n  &\\leq \\mu\\left(\\bigcup (U_n \\setminus A_n)\\right) \\\\\n  &\\leq \\sum \\mu(U_n \\setminus A_n) \\\\\n  &= \\sum (\\mu(U_n) - \\mu(A_n))  \\\\\n  &\\leq \\sum \\frac{\\varepsilon}{2^n} = \\varepsilon\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u307e\u305f, $C = \\bigcup C_n$\u3068\u304a\u304f\u3068, \u6e2c\u5ea6\u306e\u9023\u7d9a\u6027\u304b\u3089\n  \\[\n  \\mu(\\bigcup C_n) = \\lim_{N \\rightarrow \\infty} \\mu\\left(\\bigcup_{n=1}^{N} C_n\\right)\n  \\]\n  \u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, \u3042\u308b$N$\u306b\u3064\u3044\u3066$D_N = \\bigcup_{n=1}^N C_n$\u3068\u304a\u304f\u3068, $\\mu(C) - \\mu(D_N) < \\varepsilon$\u3068\u306a\u308b. \u3053\u306e\u3068\u304d, $D_N \\subset A$\u306f\u9589\u96c6\u5408\u3067\u3042\u308a, \n  \\[\n  \\begin{aligned}\n  \\mu(A) - \\mu(D_N) \n  &< \\mu(A) - \\mu(C) + \\varepsilon \\\\\n  &\\leq \\mu(\\bigcup (A_n \\setminus C_n)) + \\varepsilon \\\\\n  &\\leq \\sum (\\mu(A_n) - \\mu(C_n)) + \\varepsilon \\\\\n  &\\leq \\sum \\frac{\\varepsilon}{2^n} + \\varepsilon = 2\\varepsilon\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u4ee5\u4e0a\u304b\u3089$\\bigcup A_n \\in \\mathcal{R}$\u3068\u306a\u308b. \\\\\n  \u3055\u3066, $\\mathcal{R}$\u304c$\\sigma$-\u52a0\u6cd5\u65cf\u3067\u3042\u308b\u306e\u3067, \u3059\u3079\u3066\u306e$X$\u306e\u9589\u96c6\u5408\u304c$\\mathcal{R}$\u306b\u542b\u307e\u308c\u308b\u3053\u3068\u3092\u793a\u305b\u3070$\\mathcal{B}_X \\subset \\mathcal{R}$\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u305d\u3053\u3067, $A \\subset X$\u3092\u7a7a\u3067\u306a\u3044\u9589\u96c6\u5408\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $U_n = \\{x \\in X \\mid \\mathrm{dist}(x,A) < 1\/n\\}$\u3068\u304a\u3051\u3070, $x \\mapsto \\mathrm{dist}(x,A)$\u306e\u9023\u7d9a\u6027\u304b\u3089$U_n$\u306f\u958b\u96c6\u5408\u3067\u3042\u3063\u3066, $A$\u306f\u9589\u96c6\u5408\u3067\u3042\u308b\u304b\u3089,\n  \\[\n  \\bigcap_{n=1}^{\\infty} U_n = \\{x \\in X \\mid d(x,A) = 0 \\} = A\n  \\]\n  \u3068\u306a\u308b. \u3057\u304b\u3082, $U_1 \\supset U_2 \\supset \\cdots \\supset U_n \\supset \\cdots$\u3067\u3042\u308b\u304b\u3089, \u6e2c\u5ea6\u306e\u9023\u7d9a\u6027\u3088\u308a$\\mu(A) = \\lim_{n \\rightarrow \\infty} \\mu(U_n) = \\inf_{n} \\mu(U_n)$\u3068\u306a\u308b. \u3055\u3089\u306b$U_n \\supset A$\u3067\u3042\u308b\u304b\u3089, \n  \\[\n  \\mu(A) \n  \\leq \\inf\\{\\mu(U) \\mid U \\supset A, U\\mbox{\u306f\u958b\u96c6\u5408}\\} \n  \\leq \\inf_{n} \\mu(U_n) = \\mu(A)\n  \\]\n  \u3068\u306a\u308b. \u307e\u305f, $A$\u306f\u9589\u96c6\u5408\u306a\u306e\u3067$\\mu(A) = \\sup \\{\\mu(C) \\mid C \\subset A, C\\mbox{\u306f\u9589\u96c6\u5408}\\}$\u3068\u306a\u308b. \u3088\u3063\u3066, $A \\in \\mathcal{R}$\u3067\u3042\u308b. \n\\end{proof}\n\\begin{cor}\\label{FiniteBorelMeasOnRnIsRegular}\n  $\\mathbb{R}^n$\u4e0a\u306e\u6709\u9650Borel\u6e2c\u5ea6$\\mu$\u306f\u6b63\u5247\u3067\u3042\u308b. \u3064\u307e\u308a\u4efb\u610f\u306e$A \\in \\mathcal{B}_{\\mathbb{R}}$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  \\mu(A) \n  &= \\sup\\{\\mu(K) \\mid K \\subset A, C\\mbox{\u306f\u30b3\u30f3\u30d1\u30af\u30c8}\\} \\\\\n  &= \\inf\\{\\mu(U) \\mid U \\supset A, U\\mbox{\u306f\u958b\u96c6\u5408} \\}\n  \\end{aligned}\n  \\]\n  \u304c\u6210\u7acb\u3059\u308b. \n\\end{cor}\n\\begin{proof}\n  $\\mathbb{R}^n$\u306b\u304a\u3044\u3066\u6709\u754c\u9589\u96c6\u5408\u3068\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u306f\u4e00\u81f4\u3059\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b. $\\mathbb{R}^n$\u306f\u8ddd\u96e2\u7a7a\u9593\u3067\u3042\u308b\u306e\u3067\u3059\u3050\u4e0a\u306e\u547d\u984c\u3088\u308a\n  \\[\n  \\mu(A) = \\sup\\{\\mu(C) \\mid C \\subset A, C\\mbox{\u306f\u9589\u96c6\u5408}\\}\n  \\]\n  \u3067\u3042\u308b. \u305d\u3053\u3067, \u4efb\u610f\u306e$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, \u9589\u96c6\u5408$C \\subset A$\u3067$\\mu(A) - \\varepsilon < \\mu(C)$\u306a\u308b\u3082\u306e\u304c\u53d6\u308c\u308b. \u3053\u306e\u3068\u304d, $0 \\in \\mathbb{R}^n$\u4e2d\u5fc3\u306e\u534a\u5f84$n$\u306e\u9589\u7403\u3092$B(0;n)$\u3068\u8868\u3059\u3068, $C_n = C \\cap B(0;n)$\u306f\u6709\u754c\u9589\u96c6\u5408\u3067\u3042\u308b\u306e\u3067\u30b3\u30f3\u30d1\u30af\u30c8\u3067\u3042\u308b. \u305d\u3057\u3066, $C_1 \\subset C_2 \\subset \\cdots$\u3067\u3042\u308a, $C = \\bigcup C_n$\u3067\u3042\u308b\u306e\u3067, \u6e2c\u5ea6\u306e\u9023\u7d9a\u6027\u3088\u308a$\\mu(C)= \\lim_{n \\rightarrow \\infty  \\mu(C_n)}$\u3068\u306a\u308b. \u3057\u305f\u304c\u3063\u3066, $\\mu(C)-\\mu(C_N) < \\varepsilon$\u306a\u308b$N$\u304c\u53d6\u308c\u308b. \u3086\u3048\u306b, \n  \\[\n  \\mu(A) < \\mu(C) + \\varepsilon < \\mu(C_N) + 2\\varepsilon \\leq \\sup\\{\\mu(K) \\mid K \\subset A, C\\mbox{\u306f\u30b3\u30f3\u30d1\u30af\u30c8}\\} + 2\\varepsilon\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, $\\varepsilon > 0$\u306e\u4efb\u610f\u6027\u3088\u308a\n  \\[\n  \\mu(A) \\leq \\sup\\{\\mu(K) \\mid K \\subset A, C\\mbox{\u306f\u30b3\u30f3\u30d1\u30af\u30c8}\\}\n  \\]\n  \u3068\u306a\u308b. \u3053\u308c\u3067\u793a\u305b\u305f. \n\\end{proof}\n\n\\subsection{$C_0(\\mathbb{R}^n)$\u306e$L^p$\u7a7a\u9593\u306b\u304a\u3051\u308b\u7a20\u5bc6\u6027}\n\u672c\u7bc0\u3067\u306f, $\\mathbb{R}^n$\u4e0a\u306e\u6b63\u5247Borel\u6e2c\u5ea6$\\mu$\u3067\u3042\u3063\u3066\u4efb\u610f\u306e\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}^n$\u306b\u5bfe\u3057\u3066$\\mu(K) < \\infty$\u3068\u306a\u308b\u3082\u306e\u306b\u3064\u3044\u3066, $C_0(\\mathbb{R}^n)$\u304c$L^p(\\mathbb{R}^n,\\mathcal{B}_{\\mathbb{R}^n},\\mu)$\u306b\u304a\u3044\u3066\u7a20\u5bc6\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b. \n\n\\begin{lem}\n  $U \\subset \\mathbb{R}^n$\u3092\u958b\u96c6\u5408\u3068\u3057, $\\emptyset \\neq K \\subset U$\u3092\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u958b\u96c6\u5408$V \\subset \\mathbb{R}^n$\u3067$K \\subset V \\subset \\mathrm{cl}(V) \\subset U$\u304b\u3064$\\mathrm{cl}(V)$\u304c\u30b3\u30f3\u30d1\u30af\u30c8\u306b\u306a\u308b\u3082\u306e\u304c\u5b58\u5728\u3059\u308b. \n\\end{lem}\n\\begin{proof}\n  \u5404$x \\in U$\u306b\u5bfe\u3057\u3066, $B(x;2\\varepsilon_x) \\subset U$\u306a\u308b$\\varepsilon_x > 0$\u3092\u3068\u308b. \u3053\u306e\u3068\u304d, $K \\subset U = \\bigcup_{x \\in U} B(x;\\varepsilon_x)$\u3068\u306a\u308b. $K$\u306f\u30b3\u30f3\u30d1\u30af\u30c8\u306a\u306e\u3067\u3042\u308b$x_1,\\ldots,x_k$\u306b\u3064\u3044\u3066$K \\subset \\bigcup_{j=1}^k B(x_j;\\varepsilon_{x_j})$\u3068\u306a\u308b. \u305d\u3053\u3067, $V = \\bigcup_{j=1}^k B(x_j;\\varepsilon_{x_j})$\u3068\u304a\u3051\u3070, $V$\u306f\u958b\u96c6\u5408\u3067, $K \\subset V$\u304b\u3064, \n  \\[\n  \\mathrm{cl}(V) \\subset \\bigcup_{j=1}^k \\mathrm{cl}(B(x_j;\\varepsilon_{x_j})) \\subset \\bigcup_{j=1}^k B(x_j;2\\varepsilon_{x_j}) \\subset U\n  \\]\n  \u3067\u3042\u308b. $\\mathrm{cl}(V)$\u306e\u30b3\u30f3\u30d1\u30af\u30c8\u6027\u3082\u4e0a\u306e\u95a2\u4fc2\u304b\u3089\u308f\u304b\u308b. \n\\end{proof}\n\\begin{defn}\n  \u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b. \n  \\[\n  C_0(\\mathbb{R}^n) := \\{f \\in C(\\mathbb{R}^n) \\mid \\mathrm{supp}(f)\\mbox{\u306f\u30b3\u30f3\u30d1\u30af\u30c8}\\}.\n  \\]\n\\end{defn}\n\\begin{lem}\n  $U \\subset \\mathbb{R}^n$\u3092\u958b\u96c6\u5408\u3068\u3057, $\\emptyset \\neq K \\subset U$\u3092\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u3042\u308b$\\varphi \\in C_0(\\mathbb{R}^n)$\u304c\u5b58\u5728\u3057\u3066\u6b21\u306e$(1),(2),(3)$\u3092\u307f\u305f\u3059. \n  \\begin{itemize}\n      \\item[(1)] $\\forall x \\in K, ~ \\varphi(x) = 1$\n      \\item[(2)] $0 \\leq \\varphi \\leq1$\n      \\item[(3)] $\\mathrm{supp}\\varphi \\subset U$\n  \\end{itemize}\n\\end{lem}\n\\begin{proof}\n  $K \\subset V \\subset \\mathrm{cl}(V) \\subset U$\u304b\u3064$\\mathrm{cl}(V)$\u304c\u30b3\u30f3\u30d1\u30af\u30c8\u306b\u306a\u308b\u3088\u3046\u306a\u958b\u96c6\u5408$V \\subset \\mathbb{R}^n$\u3092\u3068\u308b. \u3053\u306e\u3068\u304d, $K \\cap (\\mathbb{R}^n \\setminus V)$\u3067\u3042\u308b\u306e\u3067, \u524d\u7bc0\u306e\u88dc\u984c\\ref{PropOfDist}\u3088\u308a, \u4efb\u610f\u306e$x \\in \\mathbb{R}^n$\u306b\u5bfe\u3057\u3066, $\\mathrm{dist}(x,K) = 0$\u304b\u3064$\\mathrm{dist}(x,\\mathbb{R}^n \\setminus V) = 0$\u3068\u306a\u308b\u3053\u3068\u306f\u306a\u3044. \u3057\u305f\u304c\u3063\u3066, $\\mathrm{dist}(x,K)+\\mathrm(x,\\mathbb{R}^n \\setminus V) > 0$\u3067\u3042\u308b. \u3053\u306e\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066, \n  \\[\n  \\varphi(x) = \\frac{\\mathrm{dist}(x,\\mathbb{R}^n \\setminus V)}{\\mathrm{dist}(x,K)+ \\mathrm{dist}(x,\\mathbb{R}^n \\setminus V)}\n  \\]\n  \u3068\u5b9a\u7fa9\u3059\u308b. \u3059\u308b\u3068, \u660e\u3089\u304b\u306b$0 \\varphi \\leq 1$\u3067\u3042\u308a, $\\mathrm{dist}$\u306e\u9023\u7d9a\u6027\u304b\u3089$\\varphi \\in C(\\mathbb{R}^n)$\u3067\u3042\u308b. \u307e\u305f, $x \\in K$\u306b\u5bfe\u3057\u3066, $\\mathrm{dist}(x,K) = 0$\u3060\u304b\u3089$\\varphi(x) = 1$\u3067\u3042\u308b. \u305d\u3057\u3066, $\\{x \\in \\mathbb{R}^n \\mid \\varphi(x) = 0\\} = \\mathbb{R}^n \\setminus V$\u3067\u3042\u308b\u306e\u3067, $\\{x \\in \\mathbb{R}^n \\mid \\varphi(x) \\neq 0\\} = V$\u3067\u3042\u308b. \u3088\u3063\u3066, $\\mathrm{supp}\\varphi = \\mathrm{cl}(V) \\subset U$\u3067\u3042\u308b. \u305d\u3057\u3066, \u3053\u306e\u95a2\u4fc2\u304b\u3089$\\mathrm{supp}\\varphi$\u306e\u30b3\u30f3\u30d1\u30af\u30c8\u6027\u304c\u308f\u304b\u308b\u306e\u3067$\\varphi \\in C_0(\\mathbb{R}^n)$\u3067\u3042\u308b. \n\\end{proof}\n\n\\begin{prop}\\label{DensityOfC0InLp}\n  $1 \\leq p < \\infty$\u3068\u3057, $\\mu$\u3092$(\\mathbb{R}^n,\\mathcal{B}_{\\mathbb{R}^n})$\u4e0a\u306e\u6b63\u5247\u6e2c\u5ea6\u3067\u4efb\u610f\u306e\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}^n$\u306b\u5bfe\u3057\u3066$\\mu(K) < \\infty$\u3068\u306a\u308b\u3082\u306e\u3068\u3059\u308b. \n  \u3053\u306e\u3068\u304d, $L^p(\\mathbb{R}^n,\\mathcal{B}_{\\mathbb{R}^n},\\mu)$\u306b\u304a\u3044\u3066$C_0(\\mathbb{R}^n)$\u306f$L^p$\u8ddd\u96e2\u3067\u7a20\u5bc6\u3067\u3042\u308b. \n  \u3059\u306a\u308f\u3061, \u4efb\u610f\u306e$f \\in L^p(\\mathbb{R}^n,\\mathcal{B}_{\\mathbb{R}^n},\\mu)$\u3068\n  $\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, $g \\in C_0(\\mathbb{R}^n)$\u304c\u5b58\u5728\u3057\u3066, \n  \\[\n  d_{\\mu}^p(f,g) = \\lVert f-g \\rVert_{L^p} =\\left(\\int_{\\mathbb{R}^r} |f - g|^p d\\mu \\right)^{1\/p} < \\varepsilon\n  \\]\n  \u3068\u306a\u308b. \n\\end{prop}\n\\begin{proof}\n  \u307e\u305a, $f = \\chi_E ~(E \\in \\mathcal{B}_{\\mathbb{R}^r})$\u306e\u5834\u5408\u3092\u8003\u3048\u308b. $f^p = f \\in L^1$\u3067\u3042\u308b\u306e\u3067, $\\mu(E) < \\infty$\u3067\u3042\u308b. \u3059\u308b\u3068, $\\mu$\u306e\u6b63\u5247\u6027\u3088\u308a$\\mu(U) < \\mu(E)+\\varepsilon$\u306a\u308b\u958b\u96c6\u5408$U \\supset E$\u3068$\\mu(E)-\\varepsilon < \\mu(K)$\u306a\u308b\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$\\emptyset \\neq K \\subset E$\u304c\u53d6\u308c\u308b. \u3044\u307e, $K \\subset U$\u306b\u5bfe\u3057\u3066, \u3059\u3050\u4e0a\u306e\u88dc\u984c\u3092\u9069\u7528\u3057\u3066$\\varphi \\in C_0(\\mathbb{R}^n)$\u3092\u53d6\u308b. \u3053\u306e\u3068\u304d, $\\mu(K) < \\infty$\u3088\u308a$\\mu(U \\setminus K) = \\mu(U) - \\mu(K) =\\mu(U) - \\mu(E) + \\mu(E) - \\mu(K) < 2\\varepsilon$\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u3068, \n  \\[\n  \\begin{aligned}\n  \\int_{\\mathbb{R}^n} |\\chi_E - \\varphi|^p d\\mu \n  &= \\int_{U} |\\chi_E - \\varphi|^p d\\mu ~~~(\\because \\mathrm{supp}\\varphi \\subset U, E \\subset U) \\\\\n  &= \\int_{U \\setminus K} |\\chi_E - \\varphi|^p d\\mu ~~~(\\because K \\subset E, \\varphi \\equiv 1 ~(\\mathrm{on} ~K)) \\\\\n  &\\leq \\int_{U \\setminus K} 2^p d\\mu = 2^p \\mu(U \\setminus K) < 2^{p+1} \\varepsilon \n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, \u3053\u306e\u5834\u5408\u306f\u6210\u7acb\u3059\u308b. \n  \u6b21\u306b$f \\geq 0$\u306e\u5834\u5408\u3092\u8003\u3048\u308b. \u3053\u306e\u3068\u304d,  \u5358\u95a2\u6570\u8fd1\u4f3c\u5b9a\u7406\u3088\u308a\u975e\u8ca0\u5358\u95a2\u6570\u5217$\\varphi_n$\u304c\u5b58\u5728\u3057\u3066, \u5404\u70b9$x$\u3067$0 \\leq \\varphi_1(x) \\leq \\varphi_2(x) \\leq \\cdots \\leq \\varphi_n(x) \\rightarrow f(x)$\u3068\u306a\u308b. \u3059\u308b\u3068, $|f - \\varphi_n|^p \\leq 2^p f^p \\in L^1$\u3067\u3042\u308b\u306e\u3067, \u512a\u53ce\u675f\u5b9a\u7406\u3088\u308a\n  \\[\n  d_{\\mu}^p(f,\\varphi_n)^p = \\int (f - \\varphi_n)^p d\\mu \\rightarrow 0\n  \\]\n  \u3068\u306a\u308b. \u305d\u3053\u3067, \u3042\u308b\u975e\u8ca0\u5358\u95a2\u6570$\\varphi = \\sum_{j=1}^n c_j \\chi_{E_j}$\u306b\u3064\u3044\u3066$d_{\\mu}^p(f,\\varphi) < \\varepsilon$\u3068\u306a\u308b. \n  \u3053\u3053\u3067, \u3059\u3050\u4e0a\u3067\u793a\u3057\u305f\u3053\u3068\u3088\u308a, \u5404$\\chi_{E_j}$\u306b\u5bfe\u3057\u3066\u9069\u5f53\u306b$g_i \\in C_0(\\mathbb{R}^r)$\u3092\u9078\u3076\u3068, $d_{\\mu}^p(\\chi_{E_j},g_i) < \\varepsilon\/n\\max_{j}\\{|c_j|\\}$\u3068\u3067\u304d\u308b. \u3086\u3048\u306b, $g= \\sum_{j=1}^n c_j g_j$\u3068\u304a\u304f\u3068, $g \\in C_0(\\mathbb{R}^r)$\u3067\u3042\u308a, \n  \\[\n  \\lVert \\varphi - g \\rVert_{L^p} \\leq \\sum_{j=1}^n |c_j| \\lVert \\chi_{E_j} - g_j \\rVert_{L^p} < \\sum_{j=1}^n |c_j| \\varepsilon\/n\\max_{1 \\leq j \\leq n}\\{|c_j|\\} \\leq \\varepsilon\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, $\\lVert f - g \\rVert_{L^p} \\leq \\lVert f - \\varphi  \\rVert_{L^p} + \\lVert \\varphi - g \\rVert_{L^p} < 2\\varepsilon$\u3068\u306a\u308b. \u4e00\u822c\u306e$f \\in L^p$\u306b\u3064\u3044\u3066\u306f$f = f^{+} - f^{-}, f^{\\pm} \\geq 0$\u3068\u5206\u89e3\u3059\u308b\u3053\u3068\u3067$f \\geq 0$\u306e\u5834\u5408\u306b\u5e30\u7740\u3055\u308c\u308b. \n\\end{proof}\n\n\u3053\u306e\u5b9a\u7406\u3092\u4f7f\u3046\u3068\u6b21\u306e\u3053\u3068\u304c\u793a\u305b\u308b. \n\\begin{prop}\\label{ContinuousOfTranslationAboutLpNorm}\n  $1 \\leq p < \\infty$\u3068\u3057, $\\mu$\u3092$(\\mathbb{R}^n,\\mathcal{B}_{\\mathbb{R}^n})$\u4e0a\u306eLebesgue\u6e2c\u5ea6\u3068\u3059\u308b. \n  \u3053\u306e\u3068\u304d, $f \\in L^p(\\mathbb{R}^n,\\mathcal{B}_{\\mathbb{R}^n},\\mu)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\lVert f(\\cdot - h) - f(\\cdot) \\rVert_{L^p(\\mu)} \\rightarrow 0 ~~~(h \\rightarrow 0).\n  \\]\n\\end{prop}\n\\begin{proof}\n  $\\varepsilon > 0$\u3092\u4efb\u610f\u306b\u53d6\u308b. \u5b9a\u7406\\ref{DensityOfC0InLp}\u3088\u308a$\\varphi \\in C_0(\\mathbb{R}^n)$\u3067$\\lVert f - \\varphi \\rVert_{L^p} < \\varepsilon$\u306a\u308b\u3082\u306e\u304c\u53d6\u308c\u308b. \n  $\\varphi$\u306f$\\mathbb{R}^n$\u4e0a\u4e00\u69d8\u9023\u7d9a\u306a\u306e\u3067\u3042\u308b$\\delta > 0$\u304c\u3042\u3063\u3066\n  \\[\n  \\forall x,y \\in \\mathbb{R}^n , |x-y|<\\delta \\Rightarrow |\\varphi(x)-\\varphi(y)|<  \\frac{\\varepsilon}{(2\\mu(\\mathrm{supp}(\\varphi))^{1\/p}} \n  \\]\n  \u3068\u306a\u308b. \u3086\u3048\u306b, \u4efb\u610f\u306e$0 < h < \\delta$\u306b\u5bfe\u3057\u3066, Lebesgue\u6e2c\u5ea6\u306e\u5e73\u884c\u79fb\u52d5\u4e0d\u5909\u6027($\\mu(E+h) = \\mu(E)$)\u3088\u308a, \n  \\[\n  \\begin{aligned}\n  \\int_{\\mathbb{R}^n} |\\varphi(x-h)-\\varphi(x)|^p d\\mu(x) \n  &= \\int_{(\\mathrm{supp}(\\varphi)+h) \\cup \\mathrm{supp}(\\varphi)} |\\varphi(x-h)-\\varphi(x)|^p d\\mu(x) \\\\\n  &\\leq \\frac{\\varepsilon^p}{2\\mu(\\mathrm{supp}(\\varphi))} (\\mu((\\mathrm{supp}(\\varphi)+h)) +  \\mu(\\mathrm{supp}(\\varphi))) \\\\\n  &= \\frac{\\varepsilon^p}{2\\mu(\\mathrm{supp}(\\varphi))} 2\\mu(\\mathrm{supp}(\\varphi)) = \\varepsilon^p\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3086\u3048\u306b, \u4efb\u610f\u306e$0 < h < \\delta$\u306b\u5bfe\u3057\u3066, Lebesgue\u6e2c\u5ea6\u306e\u5e73\u884c\u79fb\u52d5\u4e0d\u5909\u6027(\u3059\u3050\u4e0b\u306e\u6ce8\u610f)\u3088\u308a\n  \\[\n  \\begin{aligned}\n  &\\lVert f(\\cdot - h) - f(\\cdot) \\rVert_{L^p(\\mu)} \\\\\n  &\\leq \n  \\lVert f(\\cdot - h) - \\varphi(\\cdot - h) \\rVert_{L^p(\\mu)}\n  + \\lVert \\varphi(\\cdot - h) - \\varphi(\\cdot) \\rVert_{L^p(\\mu)}\n  + \\lVert \\varphi - f \\rVert_{L^p(\\mu)} \\\\\n  &\\leq 2\\lVert \\varphi - f \\rVert_{L^p(\\mu)} + \\varepsilon \\leq 2\\varepsilon\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \n\\end{proof}\n\\begin{rem}\n  \u4e0a\u306e\u8a3c\u660e\u3067\u53ef\u6e2c\u95a2\u6570$f \\geq 0$\u306b\u5bfe\u3057\u3066\n  \\[\n  \\int_{\\mathbb{R}^n} f(x-h) d\\mu(x) = \\int f(x) d\\mu(x)\n  \\]\n  \u3068\u306a\u308b\u3053\u3068\u3092\u4f7f\u3063\u305f. \u3053\u308c\u3092\u793a\u3059\u306b\u306f$f$\u306b\u53ce\u675f\u3059\u308b\u5358\u8abf\u5897\u52a0\u5358\u95a2\u6570\u5217$\\varphi_j$\u3092\u53d6\u308b(\u5358\u95a2\u6570\u8fd1\u4f3c\u5b9a\u7406). $\\varphi_j = \\sum_{k=1}^{m_j} c_{j,k} \\chi_{E_{j,k}}$\u3068\u8868\u3059\u3068\n  \\[\n  \\varphi_j(x-h) = \\sum_{k=1}^{m_j} c_{j,k} \\chi_{E_{j,k}+h}(x)\n  \\]\n  \u3068\u306a\u308b\u306e\u3067, Lebesgue\u6e2c\u5ea6\u306e\u5e73\u884c\u79fb\u52d5\u4e0d\u5909\u6027\u3088\u308a\n  \\[\n  \\int_{\\mathbb{R}^n} \\varphi_j(x-h) d\\mu(x) = \\sum_{k=1}^{m_j} c_{j,k} \\mu(E_{j,k}+h) = \\sum_{k=1}^{m_j} c_{j,k} \\mu(E_{j,k}) = \\int_{\\mathbb{R}^n}  \\varphi_j(x) d\\mu(x)\n  \\]\n  \u3068\u306a\u308b. \u3086\u3048\u306b\u5358\u8abf\u53ce\u675f\u5b9a\u7406\u3088\u308a\n  \\[\n  \\begin{aligned}\n  \\int_{\\mathbb{R}^n} f(x) d\\mu(x) \n  &= \\lim_{j \\rightarrow \\infty} \\int_{\\mathbb{R}^n} \\varphi_j(x) d\\mu(x) \\\\\n  &= \\lim_{j \\rightarrow \\infty} \\int_{\\mathbb{R}^n} \\varphi_j(x-h) d\\mu(x) \\\\\n  &= \\int_{\\mathbb{R}^n} f(x-h) d\\mu(x).\n  \\end{aligned}\n  \\]\n\\end{rem}\n\n\\subsection{Lusin\u306e\u5b9a\u7406}\n\u672c\u7bc0\u3067\u306fLusin\u306e\u5b9a\u7406\u3092\u8a3c\u660e\u3059\u308b. \u305d\u306e\u305f\u3081\u306b\u307e\u305aEgorov\u306e\u5b9a\u7406\u3092\u8a3c\u660e\u3057\u3088\u3046. \n\\begin{thm}[Egorov\u306e\u5b9a\u7406] \\\\\n  $(X,\\mathcal{M},\\mu)$\u3092\u6709\u9650\u6e2c\u5ea6\u7a7a\u9593\u3068\u3059\u308b. \u53ef\u6e2c\u95a2\u6570\u5217$(f_n:X \\rightarrow \\mathbb{R})_{n \\in \\mathbb{N}}$\u304c\u53ef\u6e2c\u95a2\u6570$f:X \\rightarrow \\mathbb{R}$\u306b\u6982\u53ce\u675f\u3059\u308b(\u3064\u307e\u308a, $f_n \\rightarrow f $ ($\\mu$-a.e.))\u306a\u3089\u3070, \u4efb\u610f\u306e$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066$E \\in \\mathcal{M}$\u304c\u5b58\u5728\u3057\u3066$\\mu(E) < \\varepsilon$\u304b\u3064$X \\setminus E$\u4e0a\u3067\u4e00\u69d8\u306b$f_n \\rightarrow f$\u3068\u306a\u308b. \n\\end{thm}\n\\begin{proof}\n  \u306f\u3058\u3081\u306b$f_n$\u304c$f$\u306b\u5404\u70b9\u53ce\u675f\u3059\u308b\u5834\u5408\u3092\u8003\u3048\u308b. $k \\in \\mathbb{N}$\u306b\u5bfe\u3057\u3066\n  \\[\n  E_n(k) := \\{ x \\in X \\mid |f_n(x)-f(x)| \\geq \\frac{1}{k} \\} ~~(n \\in \\mathbb{N})\n  \\]\n  \u3068\u304a\u304d, $F_n(k) := \\bigcup_{j=n}^{\\infty} E_j(k)$\u3068\u304a\u304f\u3068, $F_n(k) \\supset F_{n+1}(k) ~(\\forall n)$\u3067\u3042\u308a, $f_n \\rightarrow f$\u3088\u308a$\\bigcap_{n=1}^{\\infty} F_n(k) = \\emptyset$\u3067\u3042\u308b. \u5f93\u3063\u3066, $\\mu(X) < \\infty$\u306b\u6ce8\u610f\u3059\u308b\u3068, \u6e2c\u5ea6\u306e\u9023\u7d9a\u6027\u3088\u308a$\\mu(F_n(k)) \\rightarrow \\mu(\\emptyset) = 0 ~(n \\rightarrow \\infty)$\u3067\u3042\u308b. \u3086\u3048\u306b, \u4efb\u610f\u306e$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, $k \\in \\mathbb{N}$\u3054\u3068\u306b\u9069\u5f53\u306a$n_k \\in \\mathbb{N}$\u304c\u5b58\u5728\u3057\u3066$\\mu(F_{n_k}(k)) < \\varepsilon\/2^{k+1}$\u3068\u306a\u308b. \u305d\u3053\u3067$E = \\bigcup_{k=1}^{\\infty} F_{n_k}(k)$\u3068\u304a\u304f\u3068, \n  \\[\n  \\mu\\left( E \\right) = \\sum_{k=1}^{\\infty} \\mu\\left( F_{n_k}(k) \\right) \\leq \\sum_{k=1}^{\\infty} \\frac{\\varepsilon}{2^{k+1}} < \\varepsilon\n  \\]\n  \u3068\u306a\u308b. \u307e\u305f, \u4efb\u610f\u306e$\\delta > 0$\u306b\u5bfe\u3057\u3066$1\/k_0 < \\delta$\u306a\u308b$k_0 \\in \\mathbb{N}$\u3092\u53d6\u308b\u3068, \u4efb\u610f\u306e$n \\geq n_{k_0}$\u306b\u5bfe\u3057\u3066\n  \\[\n  \\forall x \\in X \\setminus E = \\bigcap_{k=1}^{\\infty} \\bigcap_{j=n_k}^{\\infty} \\left\\{x \\in X \\mid |f_j(x) - f(x)| < \\frac{1}{k} \\right\\},~ |f_n(x) - f(x) | < \\frac{1}{k_0} < \\delta\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, $X \\setminus E$\u4e0a\u3067\u4e00\u69d8\u306b$f_n \\rightarrow f$\u3068\u306a\u308b. \u6b21\u306b\u6982\u53ce\u675f\u3059\u308b\u5834\u5408\u3067\u3042\u308b\u304c, $\\mu(X \\setminus F) = 0$\u306a\u308b$F \\in \\mathcal{M}$\u4e0a\u3067\u5404\u70b9\u53ce\u675f\u3059\u308b\u3068\u3059\u308c\u3070, \u4e0a\u3067\u793a\u3057\u305f\u3053\u3068\u304b\u3089\u4efb\u610f\u306e$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066$A \\in \\mathcal{M}$\u304c\u5b58\u5728\u3057\u3066$\\mu(F\\cap A) < \\varepsilon$\u304b\u3064$F \\setminus (F\\cap A)$\u4e0a\u3067\u4e00\u69d8\u53ce\u675f\u3059\u308b. \u3088\u3063\u3066, $E = (X \\setminus F) \\cup (F \\cap A)$\u3068\u304a\u3051\u3070$\\mu(E) \\leq \\mu(X \\setminus F) + \\mu(F \\cap A) < \\varepsilon$\u304b\u3064$X \\setminus E \\subset F \\setminus (F \\cap A)$\u4e0a\u3067\u4e00\u69d8\u53ce\u675f\u3059\u308b. \n\\end{proof}\n\\begin{thm}[Lusin\u306e\u5b9a\u7406]\\label{LusinTheorem} \\\\\n  $\\mu$\u3092\u4f4d\u76f8\u7a7a\u9593$X$\u4e0a\u306e\u6709\u9650\u6b63\u5247Borel\u6e2c\u5ea6\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e\u53ef\u6e2c\u95a2\u6570$f:X \\rightarrow \\mathbb{R}$\u3068$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, \u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset X$\u304c\u5b58\u5728\u3057\u3066, $\\mu(K) > \\mu(X) - \\varepsilon$\u304b\u3064$f$\u306f$K$\u4e0a\u9023\u7d9a\u3068\u306a\u308b. \n\\end{thm}\n\\begin{proof}\n  $f \\geq 0$\u3068\u3057\u3066\u8a3c\u660e\u3059\u308c\u3070\u5341\u5206\u3067\u3042\u308b. \u5b9f\u969b, \u305d\u308c\u304c\u793a\u3055\u308c\u308c\u3070, $f = f^{+} - f^{-}$\u3068\u5206\u89e3\u3059\u308b\u3068\u304d, \u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K_{+},K_{-} \\subset X$\u304c\u5b58\u5728\u3057\u3066$\\mu(X \\setminus K_{\\pm}) < \\varepsilon\/2$\u304b\u3064$f^{\\pm}$\u306f$K_{\\pm}$\u4e0a\u9023\u7d9a\u3068\u306a\u308b\u304b\u3089, $K = K_{+} \\cap K_{-}$\u304c\u6240\u671b\u306e\u3082\u306e\u306b\u306a\u308b. \\\\\n  \u3055\u3066, $f \\geq 0$\u3068\u3059\u308b. \u307e\u305a$f$\u304c\u5358\u95a2\u6570\u3067\u3042\u308b\u5834\u5408\u3092\u8003\u3048\u308b. \u3053\u306e\u5834\u5408, $a_1,\\ldots,a_k \\geq 0$\u3068\u4e92\u3044\u306b\u7d20\u306a\u53ef\u6e2c\u96c6\u5408$E_1,\\ldots,E_k$\u306b\u3088\u308a$f = \\sum_{j=1}^k a_j \\chi_{E_j}$\u3068\u8868\u3055\u308c\u308b. \u3059\u308b\u3068,  $\\mu$\u304c\u6709\u9650\u6b63\u5247\u6e2c\u5ea6\u3067\u3042\u308b\u3053\u3068\u3088\u308a\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K_j \\subset X$\u304c\u5b58\u5728\u3057\u3066$\\mu(E_j \\setminus K_j) < \\varepsilon\/k$\u3068\u306a\u308b.  $K_1,\\ldots,K_k$\u306f\u4e92\u3044\u306b\u7d20\u3067$f$\u306f$K_j$\u4e0a\u5b9a\u6570\u3067\u3042\u308b\u304b\u3089$K = \\bigcup_{j=1}^k K_j$\u304c\u6c42\u3081\u308b\u3082\u306e\u3067\u3042\u308b. \n  \u6b21\u306b\u4e00\u822c\u306e$f \\geq 0$\u306b\u3064\u3044\u3066\u8003\u3048\u308b. \u5358\u95a2\u6570\u8fd1\u4f3c\u5b9a\u7406\u3088\u308a$f$\u306b\u53ce\u675f\u3059\u308b\u975e\u8ca0\u5024\u5358\u95a2\u6570\u5217$(\\varphi_n)_{n \\in \\mathbb{N}}$\u304c\u53d6\u308c\u308b. \u4e0a\u3067\u793a\u3057\u305f\u3053\u3068\u304b\u3089\u5404$n \\in \\mathbb{N}$\u306b\u5bfe\u3057\u3066\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K_n \\subset X$\u304c\u5b58\u5728\u3057\u3066$\\mu(X \\setminus K_n) < \\varepsilon\/2^{n+1}$\u304b\u3064$K_n$\u4e0a$\\varphi_n$\u306f\u9023\u7d9a\u3068\u306a\u308b. \u305d\u3053\u3067$K = \\bigcap_{n=1}^{\\infty} K_n$\u3068\u304a\u304f\u3068\n  \\[\n  \\mu(X \\setminus K) \\leq \\sum_{n=1}^{\\infty} \\mu(X \\setminus K_n) < \\varepsilon\/2\n  \\]\n  \u3068\u306a\u308b. Egorov\u306e\u5b9a\u7406\u3088\u308a\u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K_0 \\subset K$\u3067$\\mu(K \\setminus K_0 ) < \\varepsilon\/2$\u304b\u3064$K_0$\u4e0a\u3067\u4e00\u69d8\u306b$\\varphi_n \\rightarrow f$\u3068\u306a\u308b\u3082\u306e\u304c\u53d6\u308c\u308b. \u5f93\u3063\u3066,  \u4efb\u610f\u306e$n$\u306b\u3064\u3044\u3066$\\varphi_n$\u306f$K_0$\u4e0a\u9023\u7d9a\u3067\u3042\u308b\u306e\u3067, $f$\u306f$K_0$\u4e0a\u9023\u7d9a\u3067\u3042\u308b. \u305d\u3057\u3066, \n  \\[\n  \\mu(X \\setminus K_0) \\leq \\mu(X \\setminus K) + \\mu(K \\setminus K_0) < \\varepsilon\n  \\]\n  \u3068\u306a\u308b\u306e\u3067$K_0$\u304c\u6c42\u3081\u308b\u3082\u306e\u3067\u3042\u308b. \n\\end{proof}\n\\begin{cor}\n  $\\mu$\u3092$\\mathbb{R}^r$\u4e0a\u306e\u6709\u9650Borel\u6e2c\u5ea6\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$f \\in \\mathbb{M}^r$\u3068$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066, \u30b3\u30f3\u30d1\u30af\u30c8\u96c6\u5408$K \\subset \\mathbb{R}^r$\u304c\u5b58\u5728\u3057\u3066, $\\mu(K) > \\mu(\\mathbb{R}^r) - \\varepsilon$\u304b\u3064$K$\u4e0a\u306e\u95a2\u6570$f|K$\u306f\u9023\u7d9a\u3068\u306a\u308b. \n\\end{cor}\n\\begin{proof}\n  $\\mathbb{R}^r$\u4e0a\u306e\u6709\u9650Borel\u6e2c\u5ea6\u306f\u6b63\u5247\u3067\u3042\u308b\u304b\u3089\u4e0a\u306e\u5b9a\u7406\u3088\u308a\u5f93\u3046. \n\\end{proof}\n\n\\subsection{Clarkson\u306e\u4e0d\u7b49\u5f0f}\n\u672c\u7bc0\u3067\u306fClarkson\u306e\u4e0d\u7b49\u5f0f\u306b\u3064\u3044\u3066\u8ff0\u3079\u308b.\n\u53c2\u8003\u6587\u732e\u306f\\cite{Clarkson1936}, \\cite{MizuguchiHoshi},  \\cite{RealandAbstractAnalysis}\u3067\u3042\u308b. \n\u6b74\u53f2\u7684\u306b\u306fClarkson\u306e\u4e0d\u7b49\u5f0f\u306f$L^p$\u7a7a\u9593\u304c\u4e00\u69d8\u51f8\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059\u305f\u3081\u306b$1936$\u5e74\u306eClarkson\\cite{Clarkson1936}\u306b\u3088\u308a\u521d\u3081\u3066\u8a3c\u660e\u3055\u308c\u305f. \u307e\u305a, $0<p<\\infty$\u306e\u3068\u304d\u306e$L^p$\u7a7a\u9593\u306b\u304a\u3051\u308b\u57fa\u672c\u7684\u306a\u4e0d\u7b49\u5f0f\u306b\u3064\u3044\u3066\u8ff0\u3079\u3088\u3046. \u672c\u7bc0\u3067\u306f\u7279\u306b\u65ad\u3089\u306a\u3051\u308c\u3070$(X,\\mathcal{M},\\mu)$\u3067\u6e2c\u5ea6\u7a7a\u9593\u3092\u8868\u3059. \n\\begin{lem}\\label{LemmaForLpInequality}\n  \\begin{itemize}\n      \\item[(1)] $1 \\leq p < \\infty$\u3068\u3059\u308b\u3068, \n      \\[\n      (a+b)^p \\leq 2^{p-1}(a^p + b^p) ~~~(\\forall a,b \\geq 0).\n      \\]\n      \\item[(2)] $0 < p < 1$\u3068\u3059\u308b\u3068, \n      \\[\n      (a+b)^p \\leq a^p + b^p ~~~(\\forall a,b \\geq 0).\n      \\]\n      \\item[(3)] $1 \\leq p < \\infty, q = p\/(p-1)$\u3068\u3059\u308b\u3068, \n      \\[\n      ab \\leq \\frac{a^p}{p} + \\frac{b^q}{q} ~~~(\\forall a,b \\geq 0).\n      \\]\n      \\item[(4)] $0 < p < 1, q = p\/(p-1)$\u3068\u3059\u308b\u3068, \n      \\[\n      \\frac{a^p}{p} + \\frac{b^q}{q} \\leq ab ~~~(\\forall a \\geq 0, b > 0).\n      \\]\n  \\end{itemize}\n\\end{lem}\n\\begin{proof}\n  $(1)$: $t \\in [0,1]$\u306b\u5bfe\u3057\u3066$f(t) = (1+t)^p\/(1+t^p)$\u3068\u304a\u304f\u3068, \n  \\[\n  \\begin{aligned}\n  f'(t) \n  &= p\\frac{(1+t)^{p-1}}{(1+t^p)} - pt^{p-1}\\frac{(1+t)^p}{(1+t^p)^2} \\\\\n  &= \\frac{p(1+t)^{p-1}}{(1+t^p)^2}((1+t^p) - t^{p-1}(1+t))) \\\\\n  &= \\frac{p(1+t)^{p-1}}{(1+t^p)^2}(1-t^{p-1}) \\geq 0\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u3086\u3048\u306b$f(1) = 2^{p-1}$\u304c$f$\u306e\u6700\u5927\u5024\u3067\u3042\u308b. \u3088\u3063\u3066, \u4efb\u610f\u306e$t \\in [0,1]$\u306b\u5bfe\u3057\u3066\n  \\[\n  (1+t)^p \\leq 2^{p-1}(1+t^p)\n  \\]\n  \u3067\u3042\u308b. \u3053\u308c\u3088\u308a\u6240\u671b\u306e\u4e0d\u7b49\u5f0f\u3092\u5f97\u308b($a\/b$\u307e\u305f\u306f$b\/a$\u3092\u8003\u3048\u308c\u3070\u3088\u3044). \\\\\n  $(2)$: \u4e0a\u306e$f$\u306b\u3064\u3044\u3066$0<p<1$\u3088\u308a$f'(t) \\leq 0$\u3068\u306a\u308b\u304b\u3089$f(0) = 1$\u304c\u6700\u5927\u5024\u306b\u306a\u308b. \u3053\u308c\u3088\u308a\u5f93\u3046. \n  \\\\\n  $(3)$ $t \\in [0,\\infty)$\u306b\u5bfe\u3057\u3066$f(t) = t^{p}\/p + b^{q}\/q - bt$\u3068\u304a\u304f\u3068, \n  \\[\n  f'(t) = t^{p-1} - b\n  \\]\n  \u3067\u3042\u308b\u306e\u3067$t = b^{1\/(p-1)}$\u3067$f$\u306f\u6700\u5c0f\u5024\u3092\u3068\u308b. \n  \\[\n  f(b^{1\/(p-1)}) = \\frac{b^{p\/(p-1)}}{p} + \\frac{b^q}{q} - b^{p\/(p-1)} = \\frac{b^q}{p} + \\frac{b^q}{q} - b^q = 0\n  \\]\n  \u3067\u3042\u308b\u306e\u3067, \n  \\[\n  0 = f(b^{1\/(p-1)}) \\leq f(a) = \\frac{a^p}{p} + \\frac{b^q}{q} - ab\n  \\]\n  \u3067\u3042\u308b. \u3088\u3063\u3066\u6210\u7acb\u3059\u308b. \n  \\\\\n  $(4)$: $t \\in [0,\\infty)$\u306b\u5bfe\u3057\u3066$f(t) = t^{1\/p} - t\/p - 1\/q$\u3068\u304a\u304f\u3068, \n  \\[\n  f'(t) = \\frac{1}{p}t^{1\/p -1} - \\frac{1}{p} = \\frac{1}{p}(t^{1\/p-1}-1) \n  \\]\n  \u3067\u3042\u308b. \u3086\u3048\u306b, $1\/p-1 > 0$\u3088\u308a$0 \\leq t < 1$\u306e\u3068\u304d$f'(t) \\leq 0$\u3067\u3042\u308a\n  , $1<t<\\infty$\u306e\u3068\u304d$f'(t) \\geq 0$\u3067\u3042\u308b. \u305d\u3057\u3066$f'(1)=0$\u3067\u3042\u308b\u306e\u3067, \n  \\[\n  f(1) = 1 - 1\/p -1\/q = 0\n  \\]\n  \u304c$f$\u306e\u6700\u5c0f\u5024\u3067\u3042\u308b($1\/p + 1\/q = 1$\u306b\u6ce8\u610f). \u3088\u3063\u3066, \n  \\[\n  0 = f(1) \\leq f(a^p\/b^q) = \\frac{a}{b^{q\/p}} - \\frac{a^p}{pb^q} - \\frac{1}{q}\n  \\]\n  \u3067\u3042\u308b\u304b\u3089, \u4e21\u8fba\u306b$b^q$\u3092\u304b\u3051\u308b\u3068, \n  \\[\n  0 \\leq ab^{q-q\/p} - \\frac{a^p}{p} - \\frac{b^q}{q}\n  \\]\n  \u3068\u306a\u308b. $q\/p = q-1$\u3067\u3042\u308b\u306e\u3067\u3053\u308c\u3067\u6240\u671b\u306e\u4e0d\u7b49\u5f0f\u304c\u5f97\u3089\u308c\u305f. \n\\end{proof}\n\\begin{prop}[H\\\"{o}lder\u306e\u4e0d\u7b49\u5f0f]\\label{HolderInequality} \\\\\n  $(1)$: $1 \\leq p < \\infty$\u3068\u3057, $q = p\/(p-1)$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u53ef\u6e2c\u95a2\u6570$f, g: X \\rightarrow \\mathbb{R}$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\lVert fg \\rVert_1 \\leq \\lVert f \\rVert_p \\lVert g \\rVert_q. \n  \\]\n  \\\\\n  $(2)$: $0 < p < 1$\u3068\u3057, $q = p\/(p-1)$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $f \\in L^p(X,\\mathcal{M},\\mu), g \\in L^q(X,\\mathcal{M},\\mu)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\lVert f \\rVert_p \\lVert g \\rVert_q \\leq \\lVert fg \\rVert_1\n  \\]\n  \u304c\u6210\u308a\u7acb\u3064. \u305f\u3060\u3057, $r < 0$\u306b\u5bfe\u3057\u3066$|0|^r = +\\infty$\u3068\u89e3\u91c8\u3057, $h \\in L^r(X,\\mathcal{M},\\mu)$\u3067\u3042\u308b\u3068\u306f, $h$\u304c\u53ef\u6e2c\u95a2\u6570\u3067\u3042\u308a, \n  \\[\n  \\lVert h \\rVert_r^r := \\int_X |h|^{r} d\\mu < \\infty\n  \\]\n  \u3067\u3042\u308b\u3053\u3068\u3068\u5b9a\u7fa9\u3059\u308b(\u3053\u306e\u3068\u304d\u7279\u306b$\\mu$-a.e.\u3067$|h| > 0$\u3067\u3042\u308a, $\\lVert h \\rVert_r > 0$\u3067\u3042\u308b). \n\\end{prop}\n\\begin{proof} \\\\\n  $(1)$: $\\lVert f \\rVert_p = 0$\u306a\u3089\u3070$f=0 ~(\\mathrm{a.e.})$\u3068\u306a\u308b\u306e\u3067$\\lVert f \\rVert_p > 0$\u3068\u3057\u3066\u3088\u3044. \u540c\u69d8\u306b$\\lVert g \\rVert_q > 0$\u3068\u3057\u3066\u3088\u3044. \u3055\u3089\u306b, \u3053\u306e\u3068\u304d$f \\in L^p(X,\\mathcal{M},\\mu), g \\in L^q(X,\\mathcal{M},\\mu)$\u3068\u3057\u3066\u3088\u3044. \n  \u307e\u305a$1 < p < \\infty$\u306e\u5834\u5408\u3092\u8003\u3048\u308b. $x \\in X$\u306b\u5bfe\u3057\u3066$a = |f(x)|\/\\lVert f \\rVert_p$,$b = |g(x)|\/\\lVert g \\rVert_q$\u3068\u304a\u3044\u3066, \u3059\u3050\u4e0a\u306e\u88dc\u984c\u306e$(3)$\u3092\u9069\u7528\u3059\u308b\u3068, \n  \\[\n  \\frac{|f(x)||g(x)|}{\\lVert f \\rVert_p \\lVert g \\rVert_q} \\leq \\frac{|f(x)|^p}{p \\lVert f \\rVert_p^p} + \\frac{|g(x)|^q}{q \\lVert g \\rVert_q}\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, \u4e21\u8fba\u3092\u7a4d\u5206\u3057\u3066$1\/p + 1\/q = 1$\u3092\u4f7f\u3048\u3070\u6240\u671b\u306e\u4e0d\u7b49\u5f0f\u304c\u5f97\u3089\u308c\u308b. \n  \u6b21\u306b$p=1$\u306e\u5834\u5408\u3092\u8003\u3048\u308b. \u3053\u306e\u3068\u304d, $\\mu$-a.e.$x$\u306b\u5bfe\u3057\u3066$|f(x)g(x)| \\leq |f(x)|\\lVert g \\rVert_{\\infty}$\u304c\u6210\u308a\u7acb\u3064\u306e\u3067\u4e21\u8fba\u3092\u7a4d\u5206\u3059\u308c\u3070\u6240\u671b\u306e\u4e0d\u7b49\u5f0f\u3092\u5f97\u308b. \n  \\\\\n  $(2)$: $\\lVert f \\rVert_p > 0$\u3068\u3057\u3066\u3088\u3044. \u3053\u306e\u3068\u304d, $a = |f(x)|\/\\lVert f \\rVert_p, b = |g(x)|\/\\lVert g \\rVert_q$\u3068\u304a\u304d, \u3059\u3050\u4e0a\u306e\u88dc\u984c\u306e$(4)$\u3092\u9069\u7528\u3059\u308b\u3068, $\\mu$-a.e.$x \\in X$\u306b\u5bfe\u3057\u3066\n  \\[\n  \\frac{|f(x)|^p}{p\\lVert f \\rVert_p^p} + \\frac{|g(x)|^q}{q\\lVert g \\rVert_q^q} \n  \\leq \\frac{|f(x)||g(x)|}{\\lVert f \\rVert_p \\lVert g \\rVert_q}\n  \\]\n  \u3068\u306a\u308b. \u3057\u305f\u304c\u3063\u3066, \u4e21\u8fba\u3092\u7a4d\u5206\u3059\u308c\u3070\u6240\u671b\u306e\u4e0d\u7b49\u5f0f\u3092\u5f97\u308b. \n\\end{proof}\n\\begin{prop}[Minkowski\u306e\u4e0d\u7b49\u5f0f] \\\\\n  $0 < p < 1$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $f,g \\in L^p(X,\\mathcal{M},\\mu)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  \\lVert f + g &\\rVert_p^p \\leq \\lVert |f| + |g| \\rVert_p^p \\leq \\lVert f \\rVert_p^p + \\lVert g \\rVert_p^p, \\\\\n  &\\lVert f \\rVert_p + \\lVert g \\rVert_p \\leq \\lVert |f| + |g| \\rVert_p.\n  \\end{aligned}\n  \\]\n\\end{prop}\n\\begin{proof}\n  \u88dc\u984c\\ref{LemmaForLpInequality}$(2)$\u3088\u308a, \n  \\[\n  |f(x)+g(x)|^p \\leq (|f(x)|+|g(x)|)^p \\leq |f(x)|^p + |g(x)|^p\n  \\]\n  \u3067\u3042\u308b\u306e\u3067\u7a4d\u5206\u3059\u308b\u3053\u3068\u3067\u7b2c\u4e00\u306e\u4e0d\u7b49\u5f0f\u3092\u5f97\u308b. \u6b21\u306b\u7b2c\u4e8c\u306e\u4e0d\u7b49\u5f0f\u306b\u3064\u3044\u3066\u306fa.e.$x$\u3067$f(x),g(x) \\neq 0$\u3068\u4eee\u5b9a\u3057\u3066\u3088\u3044\u306e\u3067, a.e.$x$\u306b\u5bfe\u3057\u3066\n  \\[\n  (|f(x)| + |g(x)|)^p = (|f(x)|+|g(x)|)^{p-1}|f(x)| + (|f(x)|+|g(x)|)^{p-1}|g(x)|\n  \\]\n  \u3067\u3042\u308b\u304b\u3089, H\\\"{o}lder\u306e\u4e0d\u7b49\u5f0f\u3088\u308a\n  \\[\n  \\begin{aligned}\n  \\lVert |f| + |g| \\rVert_p^p \n  &= \\lVert (|f|+|g|)^{p-1}f \\rVert_1 + \\lVert (|f|+|g|)^{p-1}g \\rVert_1 \\\\\n  &\\geq \\lVert (|f|+|g|)^{p-1} \\rVert_{q} (\\lVert f \\rVert_p + \\lVert g \\rVert_p) \\\\\n  &= \\lVert |f| + |g| \\rVert_{p}^{p\/q} (\\lVert f \\rVert_p + \\lVert g \\rVert_p)\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u306a\u304a, $(|f|+|g|)^{p-1} \\in L^q(X,\\mathcal{M},\\mu)$\u306b\u3064\u3044\u3066\u306f\u7b2c\u4e00\u306e\u4e0d\u7b49\u5f0f\u304b\u3089\u308f\u304b\u308b. \u3088\u3063\u3066, \n  \\[\n  \\lVert |f| + |g| \\rVert_p = \\lVert |f| + |g| \\rVert_p^{p-p\/q} \\geq \\lVert f \\rVert_p + \\lVert g \\rVert_p. \n  \\]\n\\end{proof}\n\\begin{lem}\n  $1 < p \\leq 2$\u3068\u3057, $q = p\/(p-1)$\u3068\u3059\u308b\u3068, \n    \\[\n    ( 1+x )^q + (1-x)^q  \\leq 2 (1 + x^p )^{q-1} ~~~(\\forall x \\in [0,1]).\n    \\]\n\\end{lem}\n\\begin{proof}\n  \u4efb\u610f\u306b$0 < x < 1$\u3092\u53d6\u308a, \n  \\[\n  f_x(s) := (1+s^{1-q}x)(1+sx)^{q-1} + (1-s^{1-q} x)(1-sx)^{q-1} ~~~(s \\in [0,1])\n  \\]\n  \u3068\u304a\u304f. $1< p \\leq 2$\u3088\u308a$0 < x^{p-1} < 1$\u3067\u3042\u308a, $(q-1)(p-1) = 1$\u3088\u308a\n  \\[\n  f_x(x^{p-1}) = (1+x^{0})(1+x^p)^{q-1} + (1-x^0)(1-x^p)^{q-1} = 2(1+x^p)^{q-1}\n  \\]\n  \u3067\u3042\u308b. \u307e\u305f, $f_x(1) = (1+x)^q + (1-x)^q$\u3067\u3042\u308b. \u3088\u3063\u3066, $s \\in (0,1)$\u306b\u5bfe\u3057\u3066$f_x'(s) \\leq 0$\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b\u3070, \n  \\[\n  (1+x)^q + (1-x)^q = f_x(1) \\leq f_x(x^{p-1}) = 2(1+x^p)^{q-1}\n  \\]\n  \u304c\u308f\u304b\u308b. \u3055\u3066, $s \\in (0,1)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\begin{aligned}\n  f_x'(s) \n  &= (1-q)s^{-q}x(1+sx)^{q-1} + (q-1)x(1+s^{1-q}x)(1+sx)^{q-2} \\\\\n  &~~~ -(1-q)s^{-q}x(1-sx)^{q-1} - (q-1)x(1-s^{1-q}x)(1-sx)^{q-2} \\\\\n  &= (q-1)x \\left[\n  \\begin{aligned}\n  &(1+sx)^{q-2}( -s^{-q}(1+sx) +(1+s^{1-q}x) ) \\\\\n  &+ (1-sx)^{q-2} (s^{-q}(1-sx) - (1-s^{1-q}x))\n  \\end{aligned}\n  \\right] \\\\\n  &= (q-1)x(1-s^{-q})[(1+sx)^{q-2} - (1-sx)^{q-2}]\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. $1 < p \\leq 2$\u3088\u308a$2 \\leq q < \\infty$\u3067\u3042\u308b\u306e\u3067, $s \\in (0,1)$\u306b\u5bfe\u3057\u3066, $1-s^{-q} < 0$\u304b\u3064$(1+sx)^{q-2} - (1-sx)^{q-2} > 0$\u3067\u3042\u308b. \u3088\u3063\u3066, $f_x'(s) \\leq 0$\u3067\u3042\u308b. \n\\end{proof}\n\\begin{lem}\\label{ClarksonLemma}\n  \u4ee5\u4e0b\u306e\u4e0d\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3064. \n  \\begin{itemize}\n      \\item[(1)] $1 < p \\leq 2$\u3068\u3057, $q = p\/(p-1)$\u3068\u3059\u308b\u3068, \n      \\[\n       | z+w |^q + |z-w|^q  \\leq 2 (|z|^p + |w|^p )^{q-1} ~~~(\\forall z,w \\in \\mathbb{C}).\n      \\]\n      \\item[(2)] $2 \\leq p < \\infty$\u3068\u3057, $q = p\/(p-1)$\u3068\u3059\u308b\u3068, \n      \\[\n      |z+w|^q + |z-w|^q \\geq 2(|z|^p + |w|^p)^{q-1} ~~~(\\forall z,w \\in \\mathbb{C}).\n      \\]\n  \\end{itemize}\n\\end{lem}\n\\begin{proof}\n  $(1)$: $z,w=0$\u306e\u3068\u304d\u306f\u6210\u308a\u7acb\u3064\u306e\u3067$z,w \\neq 0$\u3068\u3059\u308b. $0 < |z| \\leq |w|$\u3068\u3057\u3066\u3088\u3044. \u3053\u306e\u3068\u304d, $z\/w = r \\exp(i\\theta) , 0 < r \\leq 1, 0 \\leq \\theta \\leq 2\\pi$\u3068\u8868\u305b\u308b. \u3086\u3048\u306b, $q\/(q-1) = p$\u306b\u6ce8\u610f\u3059\u308b\u3068, \u6240\u671b\u306e\u4e0d\u7b49\u5f0f\u3092\u5f97\u308b\u305f\u3081\u306b\u306f\u4efb\u610f\u306e$\\theta \\in [0,2\\pi]$\u306b\u5bfe\u3057\u3066\n  \\[\n  | 1 + r\\exp(i\\theta) |^q + |-1 + r\\exp(i\\theta) |^q\n  \\leq 2(r^p + 1)^{q-1}\n  \\]\n  \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b\u3070\u5341\u5206\u3067\u3042\u308b. $\\theta = 0$\u306e\u5834\u5408\u306f\u3059\u3050\u4e0a\u306e\u88dc\u984c\u304b\u3089\u6210\u308a\u7acb\u3064. \u305d\u306e\u307b\u304b\u306e$\\theta$\u306b\u3064\u3044\u3066\u898b\u308b\u305f\u3081\u306b, \n  \\[\n  \\begin{aligned}\n  f(\\theta) \n  :&= | 1 + r\\exp(i\\theta) |^q + |-1 + r\\exp(i\\theta) |^q \\\\\n  &= ((1+r\\cos(\\theta))^2 + r^2\\sin^2(\\theta))^{q\/2} + ((-1+r\\cos(\\theta))^2 + r^2\\sin^2(\\theta))^{q\/2} \\\\\n  &= (1+2r\\cos(\\theta) + r^2)^{q\/2} + (1-2r\\cos(\\theta) + r^2)^{q\/2}\n  \\end{aligned}\n  \\]\n  \u3068\u304a\u304f. $f(0)$\u304c$f$\u306e\u6700\u5927\u5024\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u8a3c\u660e\u306f\u5b8c\u4e86\u3059\u308b. \u5bfe\u79f0\u6027\u304b\u3089$\\theta \\in [0,\\pi\/2]$\u306e\u5834\u5408\u3092\u793a\u305b\u3070\u5341\u5206\u3067\u3042\u308b. $\\theta \\in [0,\\pi\/2]$\u306b\u5bfe\u3057\u3066\n  \\[\n  \\begin{aligned}\n  f'(\\theta) \n  &= \\frac{q}{2}(-2r \\sin(\\theta))(1+2r\\cos(\\theta) + r^2)^{\\frac{q}{2}-1} \\\\\n  &~~~ + \\frac{q}{2}(2r\\sin(\\theta))(1-2r\\cos(\\theta) + r^2)^{\\frac{q}{2}-1} \\\\\n  &= qr\\sin(\\theta)\\left( (1-2r\\cos(\\theta) + r^2)^{\\frac{q}{2}-1} - (1+2r\\cos(\\theta) + r^2)^{\\frac{q}{2}-1} \\right)\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u3055\u3089\u306b, $1 < p \\leq 2$\u3088\u308a$q \\geq 2$\u3067\u3042\u308b\u304b\u3089$\\cos(\\theta) \\geq 0$\u3088\u308a\n  \\[\n  (1-2r\\cos(\\theta) + r^2)^{\\frac{q}{2}-1} - (1+2r\\cos(\\theta) + r^2)^{\\frac{q}{2}-1} \\leq 0\n  \\]\n  \u3067\u3042\u308b. \u3088\u3063\u3066, $f'(\\theta) \\leq 0$\u3067\u3042\u308b. \u3053\u308c\u3088\u308a$f$\u306f$\\theta = 0$\u3067\u6700\u5927\u5024\u3092\u3068\u308b. \\\\\n  $(2)$: $1 < q \\leq 2$\u304b\u3064$p=q\/(q-1)$\u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, $(1)$\u3088\u308a, \n  \\[\n  | z+w |^p + |z-w|^p  \\leq 2 (|z|^q + |w|^q )^{p-1} ~~~(\\forall z,w \\in \\mathbb{C}).\n  \\]\n  \u3086\u3048\u306b, $q(p-1) = p$\u306b\u6ce8\u610f\u3059\u308b\u3068, \u4efb\u610f\u306e$u,v \\in \\mathbb{C}$\u306b\u5bfe\u3057\u3066$z=(u+v)\/2, w=(u-v)\/2$\u3068\u304a\u304f\u3053\u3068\u3067, \n  \\[\n  |u|^p + |v|^p \\leq 2 \\frac{1}{2^{p}} (|u+v|^q+|u-v|^q)^{p-1} = 2^{1-p}(|u+v|^q+|u-v|^q)^{p-1}\n  \\]\n  \u3068\u306a\u308b. $p-1=1\/(q-1)$\u3067\u3042\u308b\u304b\u3089, \n  \\[\n  2^{1\/(q-1)} (|u|^p + |v|^p) \\leq (|u+v|^q+|u-v|^q)^{1\/(q-1)}\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, \u4e21\u8fba\u3092$q-1$\u4e57\u3059\u308b\u3053\u3068\u3067\u6240\u671b\u306e\u4e0d\u7b49\u5f0f\u3092\u5f97\u308b. \n\\end{proof}\n\\begin{thm}[Clarkson\u306e\u4e0d\u7b49\u5f0f1]\\label{ClarksonInequality1} \\\\\n  $(X,\\mathcal{M},\\mu)$\u3092\u6e2c\u5ea6\u7a7a\u9593\u3068\u3059\u308b. $p \\in (1,\\infty), q = p\/(p-1)$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$f,g \\in L^p(X,\\mathcal{M},\\mu)$\u306b\u5bfe\u3057\u3066, \n  \\begin{itemize}\n      \\item[(1)] $1 < p \\leq 2$\u306a\u3089\u3070, \n      \\[\n      \\left\\lVert f+g \\right\\rVert_{p}^q + \\left\\lVert f-g \\right\\rVert_{p}^q \n      \\leq 2 \\left( \\left\\lVert f \\right\\rVert_{p}^p + \\left\\lVert g \\right\\rVert_{p}^p \\right)^{q-1}.\n      \\]\n      \\item[(2)] $2 \\leq p < \\infty$\u306a\u3089\u3070, \n      \\[\n      \\left\\lVert f+g \\right\\rVert_{p}^q + \\left\\lVert f-g \\right\\rVert_{p}^q \n      \\geq 2 \\left( \\left\\lVert f \\right\\rVert_{p}^p + \\left\\lVert g \\right\\rVert_{p}^p \\right)^{q-1}.\n      \\]\n  \\end{itemize}\n\\end{thm}\n\\begin{rem}\n  \u4e0a\u306e\u5b9a\u7406\u3067$p=2$\u306e\u5834\u5408\u306f\u5185\u7a4d\u7a7a\u9593\u306e\u4e2d\u7dda\u5b9a\u7406\u306b\u4ed6\u306a\u3089\u306a\u3044. \u3053\u308c\u3089\u306e\u4e0d\u7b49\u5f0f\u306f$L^p$\u7a7a\u9593\u304cHilbert\u7a7a\u9593\u306b\u8fd1\u3044\u6027\u8cea\u3092\u6301\u3064\u3053\u3068\u3092\u542b\u610f\u3059\u308b\\cite{Takahashi2004}. \n\\end{rem}\n\\begin{proof}\n  $(1)$: $p < 2$\u3068\u3057\u3066\u3088\u3044. $q(p-1)=p$\u306b\u6ce8\u610f\u3059\u308b. $0 < p-1 < 1$\u3067\u3042\u308b\u304b\u3089 Minkowski\u306e\u4e0d\u7b49\u5f0f\u3088\u308a, \n  \\[\n  \\begin{aligned}\n  \\lVert |f+g|^q + |f-g|^q \\rVert_{p-1}\n  &\\geq \\lVert |f+g|^q \\rVert_{p-1} + \\lVert |f-g|^q \\rVert_{p-1} \\\\\n  &=\\lVert f + g \\rVert_p^{p\/(p-1)} + \\lVert f - g \\rVert_p^{p\/(p-1)} \\\\\n  &=\\lVert f + g \\rVert_p^{q} + \\lVert f + g \\rVert_p^{q}\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u4e00\u65b9, $(q-1)(p-1)=1$\u304a\u3088\u3073\u88dc\u984c\\ref{ClarksonLemma}$(1)$\u3088\u308a, \n  \\[\n  \\begin{aligned}\n  \\lVert |f+g|^q + |f-g|^q \\rVert_{p-1}\n  &\\leq \\lVert 2(|f|^p + |g|^p)^{q-1} \\rVert_{p-1} \\\\\n  &= 2 \\left( \\int_X (|f|^p + |g|^p)^{(q-1)(p-1)} d\\mu \\right)^{1\/(p-1)} \\\\\n  &= 2 \\left( \\lVert f \\rVert_p^p + \\lVert g \\rVert_p^p \\right)^{q-1}\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u3088\u3063\u3066, \u6240\u671b\u306e\u4e0d\u7b49\u5f0f\u3092\u5f97\u308b. \\\\\n  $(2)$: $q(p-1)=p$\u306b\u6ce8\u610f\u3059\u308b. $p-1 \\geq 1$\u3067\u3042\u308b\u304b\u3089 \u4e09\u89d2\u4e0d\u7b49\u5f0f\u3088\u308a, \n  \\[\n  \\begin{aligned}\n  \\lVert |f+g|^q + |f-g|^q \\rVert_{p-1}\n  &\\leq \\lVert |f+g|^q \\rVert_{p-1} + \\lVert |f-g|^q \\rVert_{p-1} \\\\\n  &=\\lVert f + g \\rVert_p^{p\/(p-1)} + \\lVert f - g \\rVert_p^{p\/(p-1)} \\\\\n  &=\\lVert f + g \\rVert_p^{q} + \\lVert f + g \\rVert_p^{q}\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u4e00\u65b9, $(q-1)(p-1)=1$\u304a\u3088\u3073\u88dc\u984c\\ref{ClarksonLemma}$(2)$\u3088\u308a, \n  \\[\n  \\begin{aligned}\n  \\lVert |f+g|^q + |f-g|^q \\rVert_{p-1}\n  &\\geq \\lVert 2(|f|^p + |g|^p)^{q-1} \\rVert_{p-1} \\\\\n  &= 2 \\left( \\int_X (|f|^p + |g|^p)^{(q-1)(p-1)} d\\mu \\right)^{1\/(p-1)} \\\\\n  &= 2 \\left( \\lVert f \\rVert_p^p + \\lVert g \\rVert_p^p \\right)^{q-1}\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u3088\u3063\u3066, \u6240\u671b\u306e\u4e0d\u7b49\u5f0f\u3092\u5f97\u308b. \n\\end{proof}\n\u6b21\u306bClarkson\u306b\u3088\u308a\u793a\u3055\u308c\u305f\u3082\u3046\u3072\u3068\u3064\u306e\u4e0d\u7b49\u5f0f\u3092\u793a\u3057\u3066\u3044\u304f.\n\\begin{lem}\\label{ClarksonLemma2}\n  \\begin{itemize}\n      \\item[(1)] $1 < p \\leq 2$\u3068\u3059\u308b\u3068, \n      \\[\n       x^p + y^p \\geq (x^2 + y^2)^{p\/2} ~~~(\\forall x,y \\geq 0).\n      \\]\n      \\item[(2)] $2 \\leq p < \\infty$\u3068\u3059\u308b\u3068, \n      \\[\n       x^p + y^p \\leq (x^2 + y^2)^{p\/2} ~~~(\\forall x,y \\geq 0).\n      \\]\n  \\end{itemize}\n\\end{lem}\n\\begin{proof}\n  $(1)$: \u95a2\u6570$f:[0,1] \\ni t \\mapsto (t^p+1)-(1+t^2)^{p\/2} \\in \\mathbb{R}$\u304c\u975e\u8ca0\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044\u304c, $1 < p \\leq 2$\u3088\u308a$s \\mapsto s^{p\/2 -1}$\u306f\u5358\u8abf\u6e1b\u5c11\u3067\u3042\u308b\u306e\u3067, \n  \\[\n  f'(t) = pt^{p-1} - \\frac{p}{2}(2t)(1+t^2)^{p\/2 -1} \\geq pt^{p-1} - ptt^{p-1} = 0\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, $f(0) = 0$\u3068\u5408\u308f\u305b\u3066$f \\geq 0$\u3067\u3042\u308b. \\\\\n  $(2)$: $(1)$\u306e\u8a3c\u660e\u4e2d\u306e$f$\u306b\u3064\u3044\u3066, $2 \\leq p < \\infty$\u3088\u308a$s \\mapsto s^{p\/2 -1}$\u304c\u5358\u8abf\u5897\u52a0\u3067\u3042\u308b\u3053\u3068\u304b\u3089, \n  \\[\n  f'(t) = pt^{p-1} - \\frac{p}{2}(2t)(1+t^2)^{p\/2 -1} \\leq pt^{p-1} - ptt^{p-1} = 0\n  \\]\n  \u3068\u306a\u308b\u306e\u3067$f \\leq 0$\u3067\u3042\u308b. \u3053\u308c\u3088\u308a\u308f\u304b\u308b. \n\\end{proof}\n\\begin{lem}\n  \u4ee5\u4e0b\u306e\u4e0d\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3064. \n  \\begin{itemize}\n      \\item[(1)] $1 < p \\leq 2$\u3068\u3059\u308b\u3068, \n      \\[\n       2^{p-1}(|z|^p + |w|^p) \\leq | z+w |^p + |z-w|^p  \\leq 2 (|z|^p + |w|^p ) ~~~(\\forall z,w \\in \\mathbb{C}).\n      \\]\n      \\item[(2)] $2 \\leq p < \\infty$\u3068\u3059\u308b\u3068, \n      \\[\n       2^{p-1}(|z|^p + |w|^p) \\geq | z+w |^p + |z-w|^p  \\geq 2 (|z|^p + |w|^p ) ~~~(\\forall z,w \\in \\mathbb{C}).\n      \\]\n  \\end{itemize}\n\\end{lem}\n\\begin{proof}\n  $(1)$: \u7b2c\u4e00\u306e\u4e0d\u7b49\u5f0f\u306b\u3064\u3044\u3066\u306f, $1<p \\leq 2$\u3088\u308a$s \\mapsto s^{p\/2}$\u304c\u51f9\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u304a\u3088\u3073 $\\mathbb{C}$\u306b\u304a\u3051\u308b\u4e2d\u7dda\u5b9a\u7406\u3068\u88dc\u984c\\ref{ClarksonLemma2}$(1)$\u304b\u3089\n  \\[\n  \\begin{aligned}\n  |z+w|^p + |z-w|^p \n  &\\leq ( |z+w|^2 + |z-w|^2 )^{p\/2} \\\\\n  &= (2 (|z|^2 + |w|^2) )^{p\/2} \\\\\n  &= 2^{p\/2} 2^{p\/2} \\left( \\frac{|z|^2}{2} + \\frac{|w|^2}{2} \\right) \\\\\n  &\\leq 2^{p} \\frac{1}{2} (|z|^p + |w|^p)\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b\u306e\u3067\u6210\u308a\u7acb\u3064. \u7b2c\u4e8c\u306e\u4e0d\u7b49\u5f0f\u306f\u7b2c\u4e00\u306e\u4e0d\u7b49\u5f0f\u306b\u304a\u3044\u3066$z = u+v, w = u-v$\u3068\u3059\u308b\u3053\u3068\u3067\u5f97\u3089\u308c\u308b. \\\\\n  $(2)$: \u7b2c\u4e00\u306e\u4e0d\u7b49\u5f0f\u306b\u3064\u3044\u3066\u306f, $\\mathbb{C}$\u306b\u304a\u3051\u308b\u4e2d\u7dda\u5b9a\u7406\u3068\u88dc\u984c\\ref{ClarksonLemma2}$(2)$\u304a\u3088\u3073$r \\geq 1, a,b \\geq 0$\u306b\u5bfe\u3059\u308b\u4e0d\u7b49\u5f0f$(a+b)^r \\leq 2^{r-1} (a^r+b^r)$\u3088\u308a, \n  \\[\n  \\begin{aligned}\n    |z+w|^p + |z-w|^p \n    &\\leq (|z+w|^2 + |z-w|^2)^{p\/2} \\\\\n    &= ( 2(|z|^2 + |w|^2) )^{p\/2} \\\\\n    &\\leq 2^{p\/2} 2^{p\/2 -1} (|z|^p + |w|^p) \\\\\n    &= 2^{p-1} (|z|^p + |w|^p)\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b\u306e\u3067\u6210\u308a\u7acb\u3064. \u7b2c\u4e8c\u306e\u4e0d\u7b49\u5f0f\u306f\u7b2c\u4e00\u306e\u4e0d\u7b49\u5f0f\u306b\u304a\u3044\u3066$z = u+v, w = u-v$\u3068\u3059\u308b\u3053\u3068\u3067\u5f97\u3089\u308c\u308b. \n\\end{proof}\n\\begin{thm}[Clarkson\u306e\u4e0d\u7b49\u5f0f2]\\label{ClarksonInequality2} \\\\\n  $(X,\\mathcal{M},\\mu)$\u3092\u6e2c\u5ea6\u7a7a\u9593\u3068\u3059\u308b. $p \\in (1,\\infty), q = p\/(p-1)$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$f,g \\in L^p(X,\\mathcal{M},\\mu)$\u306b\u5bfe\u3057\u3066, \n  \\begin{itemize}\n      \\item[(1)] $1 < p \\leq 2$\u306a\u3089\u3070, \n      \\[\n      \\left\\lVert f+g \\right\\rVert_{p}^p + \\left\\lVert f-g \\right\\rVert_{p}^p \n      \\leq 2  (\\left\\lVert f \\right\\rVert_{p}^p + \\left\\lVert g \\right\\rVert_{p}^p ).\n      \\]\n      \\item[(2)] $2 \\leq p < \\infty$\u306a\u3089\u3070, \n      \\[\n      \\left\\lVert f+g \\right\\rVert_{p}^p + \\left\\lVert f-g \\right\\rVert_{p}^p \n      \\geq 2 \\left( \\left\\lVert f \\right\\rVert_{p}^p + \\left\\lVert g \\right\\rVert_{p}^p \\right).\n      \\]\n  \\end{itemize}\n\\end{thm}\n\\begin{proof}\n  \u524d\u88dc\u984c\u3092$f(x),g(x)$\u306b\u5bfe\u3057\u3066\u9069\u7528\u3057\u3066\u5168\u4f53\u3092\u7a4d\u5206\u3059\u308c\u3070\u3088\u3044.\n\\end{proof}\n\u6b21\u306e\u4e0d\u7b49\u5f0f\u306e\u8a3c\u660e\u306f\\cite{Sanguineti2005}\u306b\u3088\u308b. \n\\begin{thm}[Clarkson\u306e\u4e0d\u7b49\u5f0f] \\\\\n  $(X,\\mathcal{M},\\mu)$\u3092\u6e2c\u5ea6\u7a7a\u9593\u3068\u3059\u308b. $p \\in (1,\\infty), q = p\/(p-1), a = \\min\\{p,q\\}$\u3068\u3059\u308b\u3068, \u4efb\u610f\u306e$f,g \\in L^p(X,\\mathcal{M},\\mu)$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\lVert f+g \\rVert_p^a + \\lVert f-g \\rVert_p^a \\leq 2 (\\lVert f \\rVert_p^a + \\lVert g \\rVert_p^a).\n  \\]\n\\end{thm}\n\\begin{proof}\n  \u307e\u305a$1 < p \\leq 2$\u306e\u5834\u5408\u3092\u8003\u3048\u308b. \u3053\u306e\u3068\u304d, $p \\leq 2 \\leq q < \\infty$\u3067\u3042\u308b\u306e\u3067$a = \\min\\{p,q\\} = p$\u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, \u5b9a\u7406\\ref{ClarksonInequality2}$(1)$\u3088\u308a, \n  \\[\n  \\begin{aligned}\n  \\lVert f+g \\rVert_p^a + \\lVert f-g \\rVert_p^a\n  &= \\lVert f+g \\rVert_p^p + \\lVert f-g \\rVert_p^p \\\\\n  &\\leq 2(\\lVert f \\rVert_p^p + \\lVert g \\rVert_p^p) \\\\\n  &= 2 (\\lVert f \\rVert_p^a + \\lVert g \\rVert_p^a)\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u6b21\u306b$2 \\leq p < \\infty$\u306e\u5834\u5408\u3092\u8003\u3048\u308b. \u3053\u306e\u3068\u304d, $1< q \\leq 2 \\leq p$\u3067\u3042\u308b\u306e\u3067 $a = \\min\\{p,q\\} = q$\u3067\u3042\u308b. \u5b9a\u7406\\ref{ClarksonInequality1}$(2)$\u306b\u304a\u3044\u3066, $f=u+v, g=u-v$\u3068\u304a\u304f\u3053\u3068\u3067, \n  \\[\n  2^{q-1}\\left( \\lVert u \\rVert_p^q + \\lVert v \\rVert_p^q \\right) \\geq \\left( \\lVert u+v \\rVert_p^p + \\lVert u-v \\rVert_p^p \\right)^{q-1}\n  \\]\n  \u3092\u5f97\u308b. $(p-1)(q-1)=1$\u3067\u3042\u308b\u304b\u3089, \u4e21\u8fba\u3092$p-1$\u4e57\u3057\u3066\n  \\[\n  \\lVert u+v \\rVert_p^p + \\lVert u-v \\rVert_p^p \\leq 2\\left( \\lVert u \\rVert_p^q + \\lVert v \\rVert_p^q \\right)^{p-1}\n  \\]\n  \u3092\u5f97\u308b. \u4e00\u65b9, \u88dc\u984c\\ref{LemmaForLpInequality}$(1)$\u3088\u308a\u4efb\u610f\u306e$r \\in [1,\\infty), x,y \\geq 0$\u306b\u5bfe\u3057\u3066, $(x+y)^r \\leq 2^{r-1}(x^r + y^r)$\u3067\u3042\u308b\u304b\u3089, $q(p-1)=p$\u306b\u6ce8\u610f\u3059\u308b\u3068, \n  \\[\n  \\begin{aligned}\n  \\lVert u+v \\rVert_p^p + \\lVert u-v \\rVert_p^p\n  &= \\lVert u+v \\rVert_p^{q(p-1)} + \\lVert u-v \\rVert_p^{q(p-1)} \\\\\n  &\\geq \\frac{1}{2^{p-2}} \\left( \\lVert u+v \\rVert_p^q + \\lVert u-v \\rVert_p^q \\right)^{p-1}\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u3088\u3063\u3066, \n  \\[\n  \\left( \\lVert u+v \\rVert_p^q + \\lVert u-v \\rVert_p^q \\right)^{p-1} \\leq 2^{p-1} \\left( \\lVert u+v \\rVert_p^q + \\lVert u-v \\rVert_p^q \\right)^{p-1}\n  \\]\n  \u3067\u3042\u308b\u306e\u3067, $a=q$\u3088\u308a\n  \\[\n  \\lVert u+v \\rVert_p^a + \\lVert u-v \\rVert_p^a \\leq 2 \\left( \\lVert u+v \\rVert_p^a + \\lVert u-v \\rVert_p^a \\right).\n  \\]\n\\end{proof}\n\n\n\\subsection{Jackson\u8a55\u4fa1}\n\u672c\u7bc0\u3067\u306fJackson\u8a55\u4fa1\u306e\u8a3c\u660e\u3092\u4e0e\u3048\u308b. \u672c\u7bc0\u3092\u66f8\u304f\u306b\u3042\u305f\u3063\u3066\u53c2\u7167\u3057\u305f\u6587\u732e\u306f, Mhaskar\\cite{Mhaskar1995}, Mhaskar, Micchelli\\cite{MhaskarandMicchelli1995}, Schultz\\cite{Schultz}, \u732a\u72e9\\cite{igarifourier}\u3067\u3042\u308b.  \n\u307e\u305a\u6b21\u306e\u4e0d\u7b49\u5f0f\u3092\u793a\u3059. \n\\begin{thm}[\u7a4d\u5206\u5f62\u306eMinkowski\u306e\u4e0d\u7b49\u5f0f] \\\\\n  $1 \\leq p < \\infty$\u3068\u3057$(X,\\mathcal{M},\\mu),(Y,\\mathcal{N},\\nu)$\u3092$\\sigma$-\u6709\u9650\u306a\u6e2c\u5ea6\u7a7a\u9593\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u76f4\u7a4d\u53ef\u6e2c\u7a7a\u9593$X \\times Y$\u4e0a\u306e\u53ef\u6e2c\u95a2\u6570$f \\geq 0$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\left\\{ \\int_X \\left( \\int_Y f(x,y) d\\nu(y) \\right)^p d\\mu(x) \\right\\}^{1\/p} \\leq \\int_Y \\left( \\int_X f(x,y)^p d\\mu(x) \\right)^{1\/p} d\\nu(y).\n  \\]\n\\end{thm}\n\\begin{proof}\n  $p=1$\u306e\u5834\u5408\u306fFubini-Tonelli\u306e\u5b9a\u7406\u3088\u308a\u660e\u3089\u304b. $1 < p < \\infty$\u3068\u3059\u308b. \n  \\[\n  F(x) := \\int_{Y} f(x,y) d\\nu(y)\n  \\]\n  \u3068\u304a\u304f\u3068, $F$\u306f$X$\u4e0a\u306e\u53ef\u6e2c\u95a2\u6570\u3067\u3042\u308b(Fubini-Tonelli\u306e\u5b9a\u7406\u3092\u53c2\u7167\u3055\u308c\u305f\u3044). \u305d\u3057\u3066, Fubini-Tonelli\u306e\u5b9a\u7406\u3068H\\\"{o}lder\u306e\u4e0d\u7b49\u5f0f\u3088\u308a\n  \\[\n  \\begin{aligned}\n  \\int_X F(x)^p d\\mu(x)\n  &= \\int_X F(x)^{p-1} \\int_Y f(x,y) d\\nu(y) d\\mu(x) \\\\\n  &= \\int_X \\int_Y F(x)^{p-1} f(x,y) d\\nu(y) d\\mu(x) \\\\\n  &= \\int_Y \\int_X F(x)^{p-1} f(x,y) d\\mu(x) d\\nu(y) \\\\\n  &\\leq \\int_Y \\left( \\int_X F(x)^{q(p-1)} d\\mu(x) \\right)^{1\/q} \\left( \\int_X f(x,y)^p d\\mu(x) \\right)^{1\/p} d\\nu(y) \\\\\n  &= \\left( \\int_X F(x)^p d\\mu(x) \\right)^{1\/q} \\int_Y \\left( \\int_X f(x,y)^p d\\mu(x) \\right)^{1\/p} d\\nu(y)\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b\u306e\u3067$\\left( \\int_X F(x)^p d\\mu(x) \\right)^{1\/q}$\u3067\u4e21\u8fba\u3092\u5272\u308c\u3070\u6240\u671b\u306e\u4e0d\u7b49\u5f0f\u3092\u5f97\u308b. \n\\end{proof}\n\\begin{cor}\\label{CorOfMinkowskiInequality}\n  $1 \\leq p < \\infty$\u3068\u3057$(X,\\mathcal{M},\\mu),(Y,\\mathcal{N},\\nu)$\u3092$\\sigma$-\u6709\u9650\u306a\u6e2c\u5ea6\u7a7a\u9593\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $Y$\u4e0a\u306e\u53ef\u6e2c\u95a2\u6570$g \\geq 0$\u3068\u76f4\u7a4d\u53ef\u6e2c\u7a7a\u9593$X \\times Y$\u4e0a\u306e\u53ef\u6e2c\u95a2\u6570$f \\geq 0$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\left\\{ \\int_X \\left( \\int_Y g(y)f(x,y) d\\nu(y) \\right)^p d\\mu(x) \\right\\}^{1\/p} \\leq \\int_Y g(y) \\left( \\int_X f(x,y)^p d\\mu(x) \\right)^{1\/p} d\\nu(y).\n  \\]\n\\end{cor}\n\\subsubsection{\u4e00\u822cJackson\u6838\u306e\u6027\u8cea}\n\\begin{defn}[\u4e09\u89d2\u591a\u9805\u5f0f]\n  $N = 0,1,2,\\ldots$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\sum_{|\\alpha| \\leq N} c_{\\alpha} \\exp(i\\alpha^{T}x) ~~~~(c_{\\alpha} \\in \\mathbb{C},~ x \\in \\mathbb{R}^d)\n  \\]\n  \u3068\u8868\u3055\u308c\u308b\u95a2\u6570\u3092\u9ad8\u3005$N$\u6b21\u306e$d$\u5909\u6570\u4e09\u89d2\u591a\u9805\u5f0f\u3068\u3044\u3046.  \u3053\u3053\u3067$\\alpha \\in \\mathbb{Z}^d$\u3067\u3042\u308a, \n  \\[\n  |\\alpha| = |\\alpha_1| + \\cdots + |\\alpha_d|\n  \\]\n  \u3067\u3042\u308b. \u307e\u305f, $|\\alpha|=N$\u3092\u6e80\u305f\u3059\u3042\u308b$\\alpha \\in \\mathbb{Z}^d$\u306b\u5bfe\u3057\u3066$c_{\\alpha} \\neq 0$\u3068\u306a\u308b\u306a\u3089\u3070$N$\u6b21\u306e$d$\u5909\u6570\u4e09\u89d2\u591a\u9805\u5f0f\u3068\u3044\u3046. \n\\end{defn}\n\\begin{defn}[Dirichlet\u6838] \\\\\n  $N = 0,1,\\ldots $\u306b\u5bfe\u3057\u3066\u6b21\u306e\u95a2\u6570\u3092$N$\u6b21\u306eDirichlet\u6838\u3068\u3044\u3046. \n  \\[\n  D_N(x) := \\frac{\\sin\\left(N+\\frac{1}{2}\\right)}{\\sin\\frac{x}{2}} ~~~~(x \\in \\mathbb{R})\n  \\]\n\\end{defn}\n\\begin{prop}\n  \u6b21\u306e\u5f0f\u304c\u6210\u308a\u7acb\u3064. \n  \\[\n  D_N(x) = 1+2\\sum_{n=1}^N \\cos(nx) = \\sum_{n=-N}^{N} \\exp(inx) ~~~~(x \\in \\mathbb{R}).\n  \\]\n  \u7279\u306b$N$\u6b21\u306eDirichlet\u6838\u306f$N$\u6b21\u306e$1$\u5909\u6570\u4e09\u89d2\u591a\u9805\u5f0f\u3067\u3042\u308b. \n\\end{prop}\n\\begin{proof}\n  \u8907\u7d20\u6570$z \\neq 1$\u306b\u5bfe\u3057\u3066, \n  \\[\n  1+2\\sum_{n=1}^N z^n  = \\frac{1+z-2z^{n+1}}{1-z}\n  \\]\n  \u3067\u3042\u308b\u306e\u3067$z = \\exp(ix)$\u3068\u304a\u304d\u53f3\u8fba\u306e\u5206\u6bcd\u5206\u5b50\u306b$\\exp(-ix\/2)$\u3092\u304b\u3051\u308b\u3068\n  \\[\n  \\frac{\\cos(x\/2) - \\left[ \\cos\\left(N+\\frac{1}{2}\\right)x + i\\sin\\left(N+\\frac{1}{2}\\right)x \\right]}{-i\\sin(x\/2)}\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, \u5b9f\u90e8\u3092\u6bd4\u8f03\u3059\u308b\u3053\u3068\u3067\u6240\u671b\u306e\u7b49\u5f0f\u3092\u5f97\u308b. \n\\end{proof}\n\\begin{defn}[Fej\\'{e}r\u6838] \\\\\n  $N = 0,1,2,\\ldots $\u306b\u5bfe\u3057\u3066\u6b21\u306e\u95a2\u6570\u3092$N$\u6b21\u306eFej\\'{e}r\u6838\u3068\u3044\u3046. \n  \\[\n  F_N(x) := \\frac{1}{N+1}\\left[ \\frac{\\sin\\frac{1}{2}(N+1)x}{\\sin\\frac{1}{2}x} \\right]^2\n  \\]\n\\end{defn}\n\\begin{prop}\n  \u6b21\u306e\u5f0f\u304c\u6210\u308a\u7acb\u3064. \n  \\[\n  \\begin{aligned}\n  F_N(x) = 1 + 2 \\sum_{n=1}^N \\left( 1- \\frac{n}{N+1} \\right) \\cos nx = \\sum_{n=-N}^N \\left(1 - \\frac{|n|}{N+1} \\right) \\exp(i nx)\n  \\end{aligned}\n  \\]\n  \u7279\u306b$N$\u6b21\u306eFej\\'{e}r\u6838\u306f$N$\u6b21\u306e$1$\u5909\u6570\u4e09\u89d2\u591a\u9805\u5f0f\u3067\u3042\u308b. \n\\end{prop}\n\\begin{proof}\n  $D_N(x) = \\sum_{n=-N}^{N} \\exp(inx)$\u3092\u4f7f\u3046\u3068, \n  \\[\n  \\sum_{n=-N}^N \\left(1 - \\frac{|n|}{N+1} \\right) \\exp(i nx) = \\frac{1}{N+1}(D_0(x)+D_1(x)+\\cdot+D_N(x))\n  \\]\n  \u304c\u308f\u304b\u308b. \u3057\u305f\u304c\u3063\u3066, $D_N$\u306e\u5b9a\u7fa9\u3088\u308a\n  \\[\n  \\begin{aligned}\n  &\\sum_{n=-N}^N \\left(1 - \\frac{|n|}{N+1} \\right) \\exp(i nx) \\\\\n  &= \\frac{1}{N+1}\\frac{1}{2\\left(\\sin(x\/2)\\right)^2}\\left( \\sum_{n=0}^N 2\\sin(n+\\frac{1}{2})x\\cdot \\sin(x\/2) \\right) \\\\\n  &= \\frac{1}{N+1}\\frac{1}{2\\left(\\sin(x\/2)\\right)^2}\\left( \\sum_{n=0}^N (\\cos nx - \\cos(n+1)x ) \\right) \\\\\n  &= \\frac{1}{N+1}\\frac{1}{2\\left(\\sin(x\/2)\\right)^2}\\left( 1-\\cos(N+1)x \\right) \\\\\n  &= \\frac{1}{N+1}\\frac{1}{2\\left(\\sin(x\/2)\\right)^2}2 \\sin\\left( \\frac{N+1}{2}x \\right) = F_N(x)\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \n\\end{proof}\n\\begin{defn}[\u4e00\u822cJackson\u6838] \\\\\n  $N = 1,2,3,\\ldots $\u3068$r = 2,3,4,\\cdots$\u306b\u5bfe\u3057\u3066\u6b21\u306e\u95a2\u6570\u3092\u4e00\u822cJackson\u6838\u3068\u3044\u3046. \n  \\[\n  J_{N,r}(x) := \\frac{1}{c_{N,r}}\\left[ \\frac{\\sin\\frac{N}{2}x}{\\sin\\frac{1}{2}x} \\right]^{2r}.\n  \\]\n  \u3053\u3053\u306b\u5b9a\u6570$c_{N,r}$\u306f\n  \\[\n  \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} J_{N,r}(x) dx = 1\n  \\]\n  \u3068\u306a\u308b\u3088\u3046\u306b\u9078\u3076. \n\\end{defn}\n\\begin{rem}\n  \u4e00\u822cJackson\u6838\u306f$N-1$\u6b21\u306eFej\\'{e}r\u6838\u3092$r$\u4e57\u3057\u305f\u3082\u306e\u306a\u306e\u3067\u9ad8\u3005$r(N-1)$\u6b21\u306e$1$\u5909\u6570\u4e09\u89d2\u591a\u9805\u5f0f\u3067\u3042\u308b. \n\\end{rem}\n\u4e00\u822cJackson\u6838\u306e\u5b9a\u6570$c_{N,r}$\u306b\u3064\u3044\u3066\u6b21\u304c\u6210\u308a\u7acb\u3064.\n\\begin{prop}\n   $r = 2,3,4,\\cdots$\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\left( \\frac{2}{\\pi} \\right)^{2r} N^{2r-1} \\leq c_{N,r} \\leq \\pi^{2r-1} \\left( \\frac{1}{2^{2r}} + \\frac{1}{2r-1} \\right)  N^{2r-1} ~~~~(\\forall N = 0,1,2,\\ldots).\n  \\]\n\\end{prop}\n\\begin{proof}\n  \\[\n  f(x) = \\left[ \\frac{\\sin\\frac{N}{2}x}{\\sin\\frac{1}{2}x} \\right]^{2r}\n  \\]\n  \u3068\u304a\u304f\u3068\u5b9a\u7fa9\u304b\u3089\n  \\[\n  c_{N,r} = \\frac{1}{\\pi} \\int_0^\\pi f(x) dx\n  \\]\n  \u3067\u3042\u308b. $2x\/\\pi \\leq \\sin x \\leq x ~(0 \\leq x \\leq \\pi\/2)$\u3088\u308a, \n  \\[\n  \\begin{aligned}\n  c_{N,r} \n  &= \\frac{1}{\\pi} \\left( \\int_0^{1\/N} f(x) dx + \\int_{1\/N}^{\\pi} f(x) dx \\right) \\\\\n  &\\leq \\frac{1}{\\pi} \\left( \\int_0^{1\/N} \\left(\\frac{Nx\/2}{x\/\\pi} \\right)^{2r} dx + \\int_{1\/N}^{\\pi} \\left( \\frac{1}{x\/\\pi} \\right)^{2r} dx \\right) \\\\\n  &\\leq \\frac{1}{\\pi} \\left( \\left(\\frac{\\pi N}{2} \\right)^{2r} \\int_0^{1\/N} 1 dx + \\pi^{2r} \\int_{1\/N}^{\\infty} \\frac{1}{x^{2r}} dx \\right) \\\\\n  &= \\pi^{2r-1} \\left( \\frac{N^{2r-1}}{2^{2r}} + \\frac{N^{2r-1}}{2r-1} \\right)\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u307e\u305f, \n  \\[\n  \\begin{aligned}\n  c_{N,r} \n  &\\geq \\frac{1}{\\pi}\\int_{0}^{\\pi\/N} f(x) dx \\geq \\frac{1}{\\pi} \\int_0^{\\pi\/N} \\left( \\frac{\\frac{2}{\\pi}\\frac{N}{2}x}{x\/2} \\right)^{2r} dx \\\\\n  &= \\frac{1}{\\pi} \\left( \\frac{\\frac{2}{\\pi}\\frac{N}{2}x}{x\/2} \\right)^{2r} \\frac{\\pi}{N} = \\left( \\frac{2}{\\pi} \\right)^{2r} N^{2r-1}.\n  \\end{aligned}\n  \\]\n\\end{proof}\n\u3053\u306e\u547d\u984c\u3088\u308a\u4e00\u822cJackson\u6838$J_{N,r}$\u306b\u3064\u3044\u3066\u6b21\u306e\u8a55\u4fa1\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u308f\u304b\u308b. \n\\begin{prop}\\label{BoundedJacksonKernel}\n  $N=1,2,3,\\ldots$\u3068$r=2,3,4,\\ldots$\u306b\u5bfe\u3057\u3066\u6b21\u304c\u6210\u308a\u7acb\u3064.\n   \\[\n   \\begin{aligned}\n   & J_{N,r}(x) \\leq \\left( \\frac{\\pi}{2}\\right)^{4r} N ~~~~(|x| \\leq \\pi), \\\\\n   & J_{N,r}(x) \\leq \\left(\\frac{\\pi^2}{2x}\\right)^{2r} N^{-2r+1} ~~~(|x| \\leq \\pi).\n   \\end{aligned}\n   \\]\n\\end{prop}\n\\begin{proof}\n  \u7b2c\u4e00\u306e\u4e0d\u7b49\u5f0f\u306f$c_{N,r}$\u306e\u4e0b\u304b\u3089\u306e\u8a55\u4fa1\u3068$2x\/\\pi \\leq \\sin x \\leq x ~(0 \\leq x \\leq \\pi\/2)$\u3088\u308a\n  \\[\n  J_{N,r}(x) = \\frac{1}{c_{N,r}} \\left( \\frac{\\sin \\frac{N}{2}x }{\\sin \\frac{x}{2}} \\right)^{2r} \\leq \\frac{1}{N^{2r-1}} \\left( \\frac{\\pi}{2} \\right)^{2r} \\left( \\frac{\\frac{N}{2}x }{ \\frac{1}{\\pi}x} \\right)^{2r} = \\left( \\frac{\\pi}{2}\\right)^{4r} N\n  \\]\n  \u3068\u306a\u308a\u6210\u7acb\u3059\u308b. \u7b2c\u4e8c\u306e\u4e0d\u7b49\u5f0f\u306f\u540c\u69d8\u306b\n  \\[\n  J_{N,r}(x) \\leq \\frac{1}{N^{2r-1}} \\left( \\frac{\\pi}{2} \\right)^{2r} \\left( \\frac{ 1 }{ \\frac{1}{\\pi}x} \\right)^{2r} =  \\left(\\frac{\\pi^2}{2x}\\right)^{2r} N^{-2r+1}.\n  \\]\n\\end{proof}\n\\begin{lem}\\label{BoundedJacksonKernelIntegral}\n  $N = 2,3,\\ldots$\u3068$r=2,3,4,\\ldots$\u306b\u5bfe\u3057\u3066, $q=0,1,\\ldots,2r-2$\u3054\u3068\u306b\u5b9a\u6570$C_q > 0$\u304c\u5b58\u5728\u3057\u3066\n  \\[\n  \\frac{1}{\\pi} \\int_0^\\pi x^q J_{N,r}(x) dx \\leq C_q \\frac{1}{N^q}\n  \\]\n  \u3068\u306a\u308b. \u3055\u3089\u306b\u5b9a\u6570$C_q$\u3068\u3057\u3066$C_q \\leq 2^{-4r+q+2}\\pi^{4r-1}$\u3092\u6e80\u305f\u3059\u3082\u306e\u304c\u53d6\u308c\u308b. \n\\end{lem}\n\\begin{proof}\n  \u547d\u984c\\ref{BoundedJacksonKernel}\u3088\u308a\n  \\[\n  \\begin{aligned}\n  \\frac{1}{\\pi} \\int_0^\\pi x^q J_{N,r}(x) dx\n  &= \\frac{1}{\\pi} \\left( \\int_0^{2\/N} x^q J_{N,r}(x) dx + \\int_{2\/N}^\\pi x^q J_{N,r}(x) dx \\right) \\\\\n  &\\leq \\frac{1}{\\pi} \\left( \\int_0^{2\/N} x^q \\left( \\frac{\\pi}{2}\\right)^{4r} N dx + \\int_{2\/N}^\\infty x^q \\left(\\frac{\\pi^2}{2x}\\right)^{2r} N^{-2r+1} dx \\right) \\\\\n  &= \\frac{\\pi^{4r-1}}{2^{4r}} \\left( N \\int_0^{2\/N} x^q dx + 2^{2r}N^{-2r+1} \\int_{2\/N}^\\infty x^{q-2r}  dx \\right) \\\\\n  &= \\frac{\\pi^{4r-1}}{2^{4r}} \\left(  \\frac{N}{q+1} \\left( \\frac{2}{N} \\right)^{q+1} + \\frac{2^{2r}N^{-2r+1}}{2r-q-1} \\left( \\frac{2}{N} \\right)^{1+q-2r} \\right) \\\\\n  &= \\frac{\\pi^{4r-1}}{2^{4r}} \\left( \\frac{2^{q+1}}{q+1} + \\frac{2^{q+1}}{2r-q-1} \\right) \\frac{1}{N^q} \\leq \\pi^{4r-1} 2^{-4r+q+2} \\frac{1}{N^q}.\n  \\end{aligned}\n  \\]\n\\end{proof}\n\\subsubsection{\u5dee\u5206, \u6ed1\u7387\u306e\u6027\u8cea}\n\\begin{defn}[\u5dee\u5206, \u6ed1\u7387] \\\\\n  \u95a2\u6570$f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3068$r = 1,2,3,\\ldots$\u304a\u3088\u3073$h \\in \\mathbb{R}$\u306b\u5bfe\u3057\u3066\n  \\[\n  \\Delta_h^r(f)(x) := \\sum_{k=0}^r \\binom{r}{k} (-1)^{r-k} f(x+kh) ~~~~(x \\in \\mathbb{R})\n  \\]\n  \u3092$f$\u306e$r$\u968e\u306e\u5dee\u5206\u3068\u3044\u3046. \u307e\u305f, $f$\u304c$2\\pi$\u5468\u671f\u306e\u5468\u671f\u95a2\u6570\u3067$I=(-\\pi,\\pi) \\subset \\mathbb{R}$\u3068$p \\in [1,\\infty]$\u306b\u5bfe\u3057\u3066$f \\in L^p(I)$\u306e\u3068\u304d\n  \\[\n  \\omega_r(f)_{p,I}(t) := \\sup_{0 < h \\leq t} \\lVert \\Delta_h^r(f) \\rVert_{L^p(I)} ~~~~(t > 0)\n  \\]\n  \u3092$f$\u306e$r$\u968e\u306e\u6ed1\u7387(modulus of smoothness)\u3068\u3044\u3046. \n\\end{defn}\n\\begin{prop}\n  \u95a2\u6570$f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3068$h \\in \\mathbb{R}$\u306b\u5bfe\u3057\u3066\u6b21\u304c\u6210\u308a\u7acb\u3064. \n  \\[\n  \\Delta_{h}^1 (\\Delta_{h}^{r-1}(f)) =\\Delta_h^r(f) = \\Delta_h^{r-1}(\\Delta_h^1(f)) ~~~~(r = 1,2,3,\\ldots)\n  \\]\n  \u3068\u306a\u308b. \u305f\u3060\u3057, $\\Delta_h^{0}(f)=f$\u3068\u3059\u308b. \n\\end{prop}\n\\begin{proof}\n  $r$\u306b\u95a2\u3059\u308b\u5e30\u7d0d\u6cd5. $r=1$\u3067\u306e\u6210\u7acb\u306f\u660e\u3089\u304b. $r$\u3067\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068, \n  \\[\n  \\binom{r}{k} + \\binom{r}{k-1} = \\binom{r+1}{k}\n  \\]\n  \u3067\u3042\u308b\u304b\u3089, \n  \\[\n  \\begin{aligned}\n  \\Delta_h^{r}(\\Delta_h^1(f))(x)\n  &= \\sum_{k=0}^r \\binom{r}{k} (-1)^{r-k} \\Delta_h^1(f)(x+kh) \\\\\n  &= \\sum_{k=0}^r \\binom{r}{k} (-1)^{r-k} [f(x+(k+1)h)-f(x+kh)] \\\\\n  &= \\sum_{k=1}^r (-1)^{r+1-k}\\left( \\binom{r}{k} + \\binom{r}{k-1}  \\right) f(x+kh) \\\\\n  &~~~+(-1)^{r+1}f(x) + f(x+(r+1)h) \\\\\n  &= \\sum_{k=0}^{r+1} \\binom{r+1}{k} (-1)^{r+1-k} f(x+kh) = \\Delta_h^{r+1}(f)(x).\n  \\end{aligned}\n  \\]\n  \u3088\u3063\u3066, \u7b2c\u4e8c\u306e\u7b49\u5f0f\u306f\u6210\u7acb\u3059\u308b. \u540c\u69d8\u306b\n  \\[\n  \\begin{aligned}\n  \\Delta_h^1(\\Delta_h^r(f))(x) \n  &= \\Delta_h^r(f)(x+h) - \\Delta_h^r(f)(x) \\\\\n  &= \\sum_{k=0}^r \\binom{r}{k} (-1)^{r-k} f(x+(k+1)h) - \\sum_{k=0}^r \\binom{r}{k} (-1)^{r-k} f(x+kh) \\\\\n  &= \\Delta_h^{r+1}(f)(x)\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b\u304b\u3089\u7b2c\u4e00\u306e\u7b49\u5f0f\u3082\u6210\u7acb\u3059\u308b. \n\\end{proof}\n\\begin{prop}\n  \u95a2\u6570$f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3068\u6574\u6570$n \\geq 1$, $h \\in \\mathbb{R},~r=1,2,3,\\ldots$\u306b\u5bfe\u3057\u3066\u6b21\u306e\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3064. \n  \\[\n  \\Delta_{nh}^r(f)(x) = \\sum_{k_1=0}^{n-1} \\cdots \\sum_{k_r=0}^{n-1} \\Delta_h^r(f)(x+k_1 h + \\cdots + k_r h) ~~~~(x \\in \\mathbb{R}).\n  \\]\n\\end{prop}\n\\begin{proof}\n  $r$\u306b\u95a2\u3059\u308b\u5e30\u7d0d\u6cd5\u306b\u3088\u308b. \u307e\u305a, \n  \\[\n  \\sum_{k_1=0}^{n-1} \\Delta_h^1(f)(x+k_1 h) = \\sum_{k=0}^{n-1} [f(x+(k+1)h - f(x+kh)] = f(x+nh) - f(x) = \\Delta_{nh}^1(f)(x)\n  \\]\n  \u3067\u3042\u308b\u304b\u3089$r=1$\u3067\u6210\u7acb\u3059\u308b. \u6b21\u306b$r$\u3067\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068,  $\\Delta_{nh}^{r+1} = \\Delta_{nh}^1 \\Delta_{nh}^r$\u3067\u3042\u308b\u304b\u3089, \n  \\[\n  \\begin{aligned}\n  &\\Delta_{nh}^{r+1}(f)(x)  \\\\\n  &= \\Delta_{nh}^1\\left(\\sum_{k_1=0}^{n-1} \\cdots \\sum_{k_r=0}^{n-1} \\Delta_h^r(f)(\\cdot +k_1 h + \\cdots + k_r h)\\right)(x) \\\\\n  &= \\sum_{k_1=0}^{n-1} \\cdots \\sum_{k_r=0}^{n-1} \\left[\\Delta_h^r(f)(x + nh +k_1 h + \\cdots + k_r h) - \\Delta_h^r(f)(x +k_1 h + \\cdots + k_r h) \\right] \\\\\n  &= \\sum_{k_1=0}^{n-1} \\cdots \\sum_{k_r=0}^{n-1} \\sum_{k_{r+1}=0}^{n-1} \n  \\left[\n  \\begin{aligned}\n  &\\Delta_h^r(f)(x +k_1 h + \\cdots + k_r h + (k_{r+1}+1)h ) \\\\\n  &- \\Delta_h^r(f)(x +k_1 h + \\cdots + k_r h + k_{r+1} h) \n  \\end{aligned}\n  \\right] \\\\\n  &= \\sum_{k_1=0}^{n-1} \\cdots \\sum_{k_r=0}^{n-1} \\sum_{k_{r+1}=0}^{n-1} \\Delta_h^1\\left(\\Delta_h^r(f)(\\cdot +k_1 h + \\cdots + k_r h + k_{r+1}h )\\right)(x) \\\\\n  &= \\sum_{k_1=0}^{n-1} \\cdots \\sum_{k_r=0}^{n-1}  \\sum_{k_{r+1}=0}^{n-1} \\Delta_h^{r+1}(f)(x +k_1 h + \\cdots + k_r h + k_{r+1}h ).\n  \\end{aligned}\n  \\]\n\\end{proof}\n\\begin{prop}\\label{ModuliOfSmoothnessEstimate}\n  $I = (-\\pi,\\pi) \\subset \\mathbb{R}$\u3068\u304a\u304f. \u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570$f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3067\u3042\u3063\u3066, \u3042\u308b$p \\in [1,\\infty]$\u306b\u3064\u3044\u3066$f \\in L^p(I)$\u3068\u306a\u308b\u3082\u306e\u306b\u5bfe\u3057\u3066\u6b21\u306e\u4e0d\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3064. \n  \\[\n  \\omega_r(f)_{p,I}(nt) \\leq n^r \\omega_r(f)_{p,I}(t), ~~~ \\omega_r(f)_{p,I}(\\lambda t) \\leq (\\lambda+1)^r \\omega_r(f)_{p,I}(t) ~~~(\\lambda > 0).\n  \\]\n\\end{prop}\n\\begin{proof}\n  \u3059\u3050\u4e0a\u306e\u547d\u984c\u3088\u308a\n  \\[\n  \\begin{aligned}\n  w_r(f)_{p,I}(nt) \n  &= \\sup_{0 < h \\leq nt} \\lVert \\Delta_h^r(f) \\rVert_{L^p(I)} \n  = \\sup_{0 < h \\leq t} \\lVert \\Delta_{nh}^r(f) \\rVert_{L^p(I)} \\\\\n  &\\leq \\sup_{0 < h \\leq t} \\sum_{k_1=0}^{n-1} \\cdots \\sum_{k_r=0}^{n-1}  \\left\\lVert \\Delta_{h}^r(f)(\\cdot + k_1 h+\\cdots+k_r h) \\right\\rVert_{L^p(I)}\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3086\u3048\u306b$p=\\infty$\u306e\u5834\u5408\u306f$f$\u306e\u5468\u671f\u6027\u3088\u308a\u6210\u7acb\u3059\u308b. \n  \u6b21\u306b$p < \\infty$\u3068\u3059\u308b. $f$\u306e\u5468\u671f\u6027(\u304a\u3088\u3073\u3059\u3050\u4e0b\u306e\u6ce8\u610f)\u3088\u308a\n  \\[\n  \\begin{aligned}\n  &\\left\\lVert \\Delta_{h}^r(f)(\\cdot + k_1 h+\\cdots+k_r h) \\right\\rVert_{L^p(I)}^p \\\\\n  &= \\int_{-\\pi}^\\pi \\left\\lvert \\sum_{k=0}^r \\binom{r}{k} (-1)^{r-k} f(x+kh+k_1 h + \\cdots + k_r h) \\right\\rvert^p dx \\\\\n  &= \\int_{-\\pi + k_1 h + \\cdots + k_r h}^{\\pi + k_1 h + \\cdots + k_r h} \\left\\lvert \\sum_{k=0}^r \\binom{r}{k} (-1)^{r-k} f(x+kh) \\right\\rvert^p dx \\\\\n  &= \\int_{-\\pi}^{\\pi} \\left\\lvert \\sum_{k=0}^r \\binom{r}{k} (-1)^{r-k} f(x+kh) \\right\\rvert^p dx = \\left\\lVert \\Delta_h^r(f) \\right\\rVert_{L^p(I)}^p.\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u3088\u3063\u3066, \n  \\[\n  \\begin{aligned}\n  w_r(f)_{p,I}(nt) \n  &\\leq \\sup_{0 < h \\leq t} \\sum_{k_1=0}^{n-1} \\cdots \\sum_{k_r=0}^{n-1}  \\left\\lVert \\Delta_{h}^r(f)(\\cdot + k_1 h+\\cdots+k_r h) \\right\\rVert_{L^p(I)} \\\\\n  &= n^r \\sup_{0 < h \\leq t} \\left\\lVert \\Delta_h^r(f) \\right\\rVert_{L^p(I)} = n^r \\omega_r(f)_{p,I}(t).\n  \\end{aligned}\n  \\]\n  \u7b2c\u4e8c\u5f0f\u306b\u3064\u3044\u3066\u306f$n < \\lambda \\leq n+1$\u306a\u308b\u6574\u6570$n$\u3092\u3068\u308b\u3068, \u524d\u534a\u3088\u308a\n  \\[\n  w_r(f)_{p,I}(\\lambda t) \\leq w_r(f)_{p,I}((n+1)t) \\leq (n+1)^r \\omega_r(f)_{p,I}(t) \\leq (\\lambda+1)^r \\omega_r(f)_{p,I}(t).\n  \\]\n\\end{proof}\n\\begin{rem}\n  \u5468\u671f$2a$\u306e\u95a2\u6570$f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3068$b \\in \\mathbb{R}$\u306b\u5bfe\u3057\u3066\n  \\[\n  \\int_{-a+b}^{a+b} f(x) dx = \\int_{-a}^a f(x) dx\n  \\]\n  \u3067\u3042\u308b. \u5b9f\u969b, \n  \\[\n  \\begin{aligned}\n  \\int_{-a+b}^{a+b} f(x) dx \n  &= \\int_{-a+b}^{a} f(x) dx + \\int_{a}^{a+b} f(x) dx \\\\\n  &= \\int_{-a+b}^{a} f(x) dx + \\int_{-a}^{-a+b} f(x+2a) dx \\\\\n  &= \\int_{-a+b}^{a} f(x) dx + \\int_{-a}^{-a+b} f(x) dx = \\int_{-a}^a f(x) dx.\n  \\end{aligned}\n  \\]\n\\end{rem}\n\\begin{prop}\n  $I= (-\\pi,\\pi)$\u3068\u304a\u304f. \u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570$f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3067\u3042\u3063\u3066, \u3042\u308b$p \\in [1,\\infty]$\u306b\u3064\u3044\u3066$f \\in L^p(I)$\u3067\u3042\u308b\u3082\u306e\u306b\u5bfe\u3057\u3066, \n  \\[\n  \\lVert \\Delta_t^r(f) \\rVert_{L^p(I)} \\leq w_r(f)_{p,I}(|t|) ~~~~(|t| < \\pi). \n  \\]\n\\end{prop}\n\\begin{proof}\n  $t \\geq 0$\u306e\u3068\u304d\u306f\u5b9a\u7fa9\u3088\u308a\u660e\u3089\u304b\u3067\u3042\u308b. $t \\in (-\\pi,0)$\u3068\u3059\u308b.\n  $f$\u306e\u5468\u671f\u6027\u3088\u308a\n  \\[\n  \\begin{aligned}\n  \\lVert \\Delta_t^r(f) \\rVert_{L^p(I)} \n  &= \\left\\lVert  \\sum_{k=0}^r \\binom{r}{k} (-1)^{r-k} f(\\cdot+kt)  \\right\\rVert_{L^p(I)} \\\\\n  &= \\left\\lVert  \\sum_{k=0}^r \\binom{r}{k} (-1)^{r-k} f(\\cdot+(k-r)t) \\right\\rVert_{L^p(I)} \\\\\n  &= \\left\\lVert \\sum_{k=0}^r \\binom{r}{r-k} (-1)^{k} f(x+k(-t)) \\right\\rVert_{L^p(I)} \\\\\n  &= \\left\\lVert (-1)^{-r} \\sum_{k=0}^r \\binom{r}{k} (-1)^{r-k} f(\\cdot+k(-t)) \\right\\rVert_{L^p(I)} \\\\\n  &= \\lVert \\Delta_{-t}^r(f) \\rVert_{L^p(I)} \\leq w_r(f)_{p,I}(|t|).\n  \\end{aligned}\n  \\]\n\\end{proof}\n\\subsubsection{$L^p$\u304b\u3064$2\\pi$\u5468\u671f\u3067$1$\u5909\u6570\u306e\u5834\u5408}\n\\begin{lem}\n  \u95a2\u6570$g:\\mathbb{R} \\rightarrow \\mathbb{R}$\u306f\u3042\u308b\u6b63\u306e\u6574\u6570$k$\u306b\u3064\u3044\u3066$2\\pi\/k$\u5468\u671f\u306e\u95a2\u6570\u3067\u3042\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $k$\u3067\u5272\u308a\u5207\u308c\u306a\u3044\u4efb\u610f\u306e\u6574\u6570$l$\u306b\u3064\u3044\u3066\n  \\[\n  \\int_{-\\pi}^\\pi g(t) \\cos(lt) dt = \\int_{-\\pi}^\\pi g(t) \\sin(lt) dt = 0. \n  \\]\n\\end{lem}\n\\begin{proof}\n  $g$\u306f$2\\pi$\u5468\u671f\u95a2\u6570\u3067\u3082\u3042\u308b\u306e\u3067, \n  \\[\n  \\begin{aligned}\n  \\int_{-\\pi}^\\pi g(t) \\exp(ilt) dt \n  &= \\int_{-\\pi+2\\pi\/k}^{\\pi+2\\pi\/k} g(s+2\\pi\/k) \\exp(il(s+2\\pi\/k)) ds \\\\\n  &= \\exp(2\\pi il\/k) \\int_{-\\pi+2\\pi\/k}^{\\pi+2\\pi\/k} g(s) \\exp(ils) ds \\\\\n  &= \\exp(2\\pi il\/k) \\int_{-\\pi}^{\\pi} g(s) \\exp(ils) ds\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. $\\exp(2\\pi il\/k) \\neq 1$\u3067\u3042\u308b\u306e\u3067\u6240\u671b\u306e\u7b49\u5f0f\u3092\u5f97\u308b. \n\\end{proof}\n\\begin{lem}\\label{DefApproOperater}\n  $r \\geq 2$\u3068\u3059\u308b. \u6574\u6570$n \\geq 1$\u306b\u5bfe\u3057\u3066$m(n,r)= \\lfloor n\/r \\rfloor + 1$\u3068\u304a\u304d$K_{n,r} := J_{m(n,r), r}$\u3068\u304a\u304f(\u305f\u3060\u3057, $\\lfloor x \\rfloor$\u306f$x$\u3092\u8d85\u3048\u306a\u3044\u6700\u5927\u306e\u6574\u6570\u3092\u8868\u3057, $J_{N,r}$\u306f\u4e00\u822cJackson\u6838\u3067\u3042\u308b). \u3059\u308b\u3068, \u5b9a\u6570$C_r > 0$\u304c\u5b58\u5728\u3057\u3066\n  \\[\n  \\int_0^\\pi t^q K_{n,r}(t) dt \\leq C_r \\frac{1}{n^q} ~~~~(q=0,1,\\ldots,2r-2)\n  \\]\n  \u3068\u306a\u308b. \u307e\u305f, \u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570$f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3067$f \\in L^1(I)$\u306a\u308b\u3082\u306e\u306b\u5bfe\u3057\u3066\n  \\[\n  S_{n,r}(f)(x) := \\int_{-\\pi}^\\pi \\left( \\Delta_t^r(f)(x) + f(x) \\right) K_{n,r}(t) dt\n  \\]\n  \u3068\u5b9a\u3081\u308b\u3068, $S_{n,r}(f)$\u306f\u9ad8\u3005$n$\u6b21\u306e\u4e09\u89d2\u591a\u9805\u5f0f\u3067\u3042\u308b. \n  \n\\end{lem}\n\\begin{proof}\n  (\u524d\u534a):\u88dc\u984c\\ref{BoundedJacksonKernelIntegral}\u3088\u308a\u5b9a\u6570$C_r > 0$\u304c\u5b58\u5728\u3057\u3066$q = 0,1,\\ldots,2r-2$\u306b\u5bfe\u3057\u3066\n  \\[\n  \\int_0^\\pi t^q K_{n,r}(t) dt = \\int_0^{\\pi} t^q J_{m(n,r),r}(t) dt \\leq C_r \\frac{1}{m(n,r)^q} \\leq C_r \\frac{r^q}{n^q}\n  \\]\n  \u3068\u306a\u308b. \u3057\u305f\u304c\u3063\u3066, \u6539\u3081\u3066$C_r r^{2r-2}$\u3092$C_r$\u3068\u304a\u304f\u3053\u3068\u306b\u3059\u308c\u3070\n  \\[\n  \\int_0^\\pi t^q K_{n,r}(t) dt \\leq C_r \\frac{1}{n^q} ~~~~(q=0,1,\\ldots,2r-2)\n  \\]\n  \u3068\u306a\u308b. \\\\\n  (\u5f8c\u534a):\u4e00\u822cJackson\u6838\u306e\u5b9a\u7fa9\u3088\u308a$K_{n,r} = J_{m(n,r),r}$\u306f\u9ad8\u3005$r ( m(n,r)-1) = r \\lfloor n\/r \\rfloor \\leq n$\u6b21\u306e\u4e09\u89d2\u591a\u9805\u5f0f\u3067\u3042\u308a, \u5076\u95a2\u6570\u3067\u3042\u308b\u306e\u3067$K_{n,r}(t)$\u306f$1,\\cos(t),\\cos(2t),\\ldots,\\cos(nt)$\u306e\u7dda\u5f62\u7d50\u5408\u3067\u8868\u3055\u308c\u308b. \u3086\u3048\u306b$S_{n,r}(f)(x)$\u306f\n  \\[\n  \\int_{-\\pi}^{\\pi} f(x+kt) \\cos(lt) dt ~~~~(k=0,\\ldots,r,~l=1,\\ldots,n)\n  \\]\n  \u306e\u7dda\u5f62\u7d50\u5408\u3067\u8868\u3055\u308c\u308b. $k=0$\u306e\u3068\u304d\u4e0a\u306e\u5f0f\u306f\u5b9a\u6570\u3067\u3042\u308b. \n  $k=1,\\ldots,r$\u306b\u5bfe\u3057\u3066\u306f$t \\mapsto f(x+kt)$\u306f$2\\pi\/k$\u5468\u671f\u306a\u306e\u3067\u3059\u3050\u4e0a\u306e\u88dc\u984c\u3088\u308a$k$\u3067\u5272\u308a\u5207\u308c\u306a\u3044\u4efb\u610f\u306e$l=1,\\ldots,n$\u306b\u3064\u3044\u3066\n  \\[\n  \\int_{-\\pi}^{\\pi} f(x+kt) \\cos(lt) dt = 0\n  \\]\n  \u3068\u306a\u308b. \u307e\u305f, $k$\u3067\u5272\u308a\u5207\u308c\u308b\u4efb\u610f\u306e$l=1,\\ldots,n$\u306b\u3064\u3044\u3066\u306f$m=l\/k$\u3068\u304a\u304f\u3068\n  \\[\n  \\begin{aligned}\n  &\\int_{-\\pi}^{\\pi} f(x+kt) \\cos(lt) dt \\\\\n  &= \\int_{x-k\\pi}^{x+k\\pi} f(u) \\cos\\left( \\frac{l(u-x)}{k} \\right) \\frac{1}{k} du ~~~~(u = x+kt)\\\\\n  &= \\int_{x-k\\pi}^{x+k\\pi} \\frac{1}{k} f(u) (\\cos(mu)\\cos(mx) + \\sin(mu)\\sin(mx)) du \\\\\n  &= \\left(\\int_{x-k\\pi}^{x+k\\pi} \\frac{1}{k} f(u) \\cos(mu) du \\right)\\cos(mx)  + \\left(\\int_{x-k\\pi}^{x+k\\pi} \\frac{1}{k} f(u) \\sin(mu) du \\right) \\sin(mx) \\\\\n  &= \\left(\\int_{-k\\pi}^{k\\pi} \\frac{1}{k} f(u) \\cos(mu) du \\right)\\cos(mx) + \\left(\\int_{-k\\pi}^{k\\pi} \\frac{1}{k} f(u) \\sin(mu) du \\right) \\sin(mx)\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, $m \\leq n$\u306b\u6ce8\u610f\u3059\u308c\u3070$S_{n,r}(f)$\u304c\u9ad8\u3005$n$\u6b21\u306e\u4e09\u89d2\u591a\u9805\u5f0f\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b.\n\\end{proof}\n\\begin{thm}\\label{JacksonEstimate0}\n  $I= (-\\pi,\\pi)$\u3068\u304a\u304d, $r \\geq 2$, $p \\in [1,\\infty]$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u6b21\u3092\u307f\u305f\u3059\u5b9a\u6570$C$\u304c\u5b58\u5728\u3059\u308b: \u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570$f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3067\u3042\u3063\u3066, \u3042\u308b\u306b\u3064\u3044\u3066$f \\in L^p(I)$\u3067\u3042\u308b\u3082\u306e\u306b\u5bfe\u3057\u3066\n  \\[\n  \\lVert S_{n,r}(f) - f \\rVert_{L^p(I)} \\leq C \\omega_r(f)_{p,I}\\left(\\frac{1}{n}\\right) ~~~~(n=1,2,3,\\ldots)\n  \\]\n  \u3068\u306a\u308b. \n\\end{thm}\n\\begin{proof}\n  \u3059\u3050\u4e0a\u306e\u88dc\u984c\u306e\u524d\u534a\u3088\u308a, $n$\u306b\u4f9d\u5b58\u3057\u306a\u3044\u5b9a\u6570$C$\u304c\u5b58\u5728\u3057\u3066\n  \\[\n  \\int_{-\\pi}^{\\pi} K_{n,r}(t) (n|t|+1)^r dt = 2 \\int_0^\\pi K_{n,r}(t) (n|t|+1)^r \\leq C\n  \\]\n  \u3068\u306a\u308b. \u307e\u305a$p=\\infty$\u306e\u5834\u5408\u3092\u8003\u3048\u308b. \n  \\[\n  \\int_{-\\pi}^\\pi K_{n,r}(t) dt = 1\n  \\]\n  \u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308c\u3070, \u4efb\u610f\u306e$x \\in I$\u306b\u5bfe\u3057\u3066\u547d\u984c\\ref{ModuliOfSmoothnessEstimate}\u3088\u308a\n  \\[\n  \\begin{aligned}\n  |S_{n,r}(f)(x) - f(x)|\n  &= \\left\\lvert \\int_{-\\pi}^\\pi \\left[ \\Delta_t^r(f)(x) + f(x) \\right] K_{n,r}(t) dt -  \\int_{-\\pi}^\\pi f(x) K_{n,r}(t) dt \\right\\rvert \\\\\n  &= \\left\\lvert \\int_{-\\pi}^\\pi \\Delta_t^r(f)(x) K_{n,r}(t) dt \\right\\rvert \\leq \\int_{-\\pi}^\\pi \\lvert \\Delta_t^r(f)(x) K_{n,r}(t) \\rvert dt \\\\\n  &\\leq \\int_{-\\pi}^\\pi \\lVert \\Delta_t^r(f)(x) \\rVert_{L^\\infty(I)} K_{n,r}(t) dt \\leq \\int_{-\\pi}^{\\pi} K_{n,r}(t) \\omega_r(f)_{\\infty,I}(|t|) dt \\\\\n  &\\leq \\int_{-\\pi}^{\\pi} K_{n,r}(t) \\omega_r(f)_{\\infty,I}\\left(n|t|\\frac{1}{n}\\right) dt \\\\\n  &\\leq \\omega_r(f)_{\\infty,I}\\left(\\frac{1}{n}\\right) \\int_{-\\pi}^{\\pi} K_{n,r}(t) (n|t|+1)^r dt \\leq C \\omega_r(f)_{\\infty,I}\\left(\\frac{1}{n}\\right).\n  \\end{aligned}\n  \\]\n  \u6b21\u306b$1 \\leq p < \\infty$\u306e\u5834\u5408\u3092\u8003\u3048\u308b. \u7a4d\u5206\u5f62\u306eMinkowski\u306e\u4e0d\u7b49\u5f0f\u304a\u3088\u3073\u547d\u984c\\ref{ModuliOfSmoothnessEstimate}\u3088\u308a\n  \\[\n  \\begin{aligned}\n  &\\lVert S_{n,r}(f) - f \\rVert_{L^p(I)} \\\\\n  &= \\left( \\int_{-\\pi}^{\\pi} \\left|\\int_{-\\pi}^\\pi  \\Delta_t^r(f)(x) K_{n,r}(t) dt \\right|^p dx \\right)^{1\/p} \\\\\n  &\\leq \\int_{-\\pi}^{\\pi} K_{n,r}(t) \\lVert \\Delta_t^r(f) \\rVert_{L^p(I)} dt \\leq \\int_{-\\pi}^{\\pi} K_{n,r}(t) \\omega_r(f)_{p,I}(|t|) dt \\\\\n  &\\leq \\omega_r(f)_{p,I}\\left(\\frac{1}{n}\\right) \\int_{-\\pi}^{\\pi} K_{n,r}(t) (n|t|+1)^r dt \\leq C \\omega_r(f)_{p,I}\\left(\\frac{1}{n}\\right).\n  \\end{aligned} \n  \\]\n\\end{proof}\n\n\\subsubsection{$L^p$\u304b\u3064$2\\pi$\u5468\u671f\u3067\u591a\u5909\u6570\u306e\u5834\u5408}\n\u3053\u306e\u7bc0\u3067\u306f\u591a\u5909\u6570\u306e\u5834\u5408\u3092\u6271\u3046. \n\n\\begin{lem}\n  $I = (-\\pi,\\pi)$\u3068\u304a\u304d, $1 \\leq p \\leq \\infty$\u3068\u3059\u308b. \u307e\u305f, $T$\u3092$L^p(I)$\u304b\u3089$L^p(I)$\u3078\u306e\u6709\u754c\u7dda\u5f62\u5199\u50cf\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $f \\in L^p(I^d)$\u306b\u5bfe\u3057\u3066$f$\u306e$i$\u756a\u76ee\u306e\u5909\u6570\u4ee5\u5916\u3092\u56fa\u5b9a\u3057\u3066\u5f97\u3089\u308c\u308b$1$\u5909\u6570\u95a2\u6570\u306b$T$\u3092\u4f5c\u7528\u3055\u305b\u305f\u3082\u306e\u3092$T_i f$\u3068\u304b\u304f\u3053\u3068\u306b\u3059\u308b. \u3064\u307e\u308a, $x_1,\\ldots,x_{i-1},x_i,x_{i+1},\\ldots,x_d \\in I$\u306b\u5bfe\u3057\u3066\u95a2\u6570$f_i$\u3092\n  \\[\n  f_i:I \\ni x \\rightarrow f(x_1,\\ldots,x_{i-1},x,x_{i+1},\\ldots,x_d) \\in \\mathbb{R}\n  \\]\n  \u306b\u3088\u308a\u5b9a\u3081$(T_i f)(x_1,\\ldots,x_d) := (T f_i)(x_i)$\u3068\u5b9a\u7fa9\u3059\u308b. \u3059\u308b\u3068$T_i f \\in L^p(I^d)$\u3067\u3042\u308b. \n\\end{lem}\n\\begin{proof}\n  \\[\n  \\lVert T_i f \\rVert_{L^p(I^d)} = \\lVert \\lVert T_i f \\rVert_{L^p(I)} \\rVert_{L^p(I^{d-1})} \\leq \\lVert \\lVert T \\rVert \\lVert f_i \\rVert_{L^p(I)} \\rVert_{L^p(I^{d-1})} = \\lVert T \\rVert \\lVert f \\rVert_{L^p(I^d)}.\n  \\]\n\\end{proof}\n\\begin{lem}\\label{LemforMultiJacksonEst}\n  $I = (-\\pi,\\pi)$\u3068\u304a\u304d, $1 \\leq p \\leq \\infty$\u3068\u3059\u308b. \u307e\u305f, $T$\u3092$L^p(I)$\u304b\u3089$L^p(I)$\u3078\u306e\u6709\u754c\u7dda\u5f62\u5199\u50cf\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $f \\in L^p(I^d)$\u306b\u5bfe\u3057\u3066, $\\varepsilon_i := \\lVert f - T_i f  \\rVert_{L^p(I^d)}$\u3068\u304a\u304d, $N = 1,\\ldots,d$\u306b\u5bfe\u3057\u3066$T^N = T_N T_{N-1} \\cdots T_1$\u3068\u304a\u304f\u3068, \n  \\[\n  \\lVert f - T^N f \\rVert_{L^p(I^d)} \\leq \\sum_{i=1}^N  \\varepsilon_i \\lVert T \\rVert^{N-i} .\n  \\]\n\\end{lem}\n\\begin{proof}\n  $N$\u306b\u95a2\u3059\u308b\u5e30\u7d0d\u6cd5. $N=1$\u3067\u306e\u6210\u7acb\u306f\u660e\u3089\u304b. $N$\u3067\u6210\u7acb\u3059\u308b\u3068\u3059\u308b\u3068, \n  \\[\n  \\begin{aligned}\n  \\lVert f - T^{N+1} f \\rVert_{L^p(I^d)}\n  &\\leq \\lVert f -  T_{N+1} f \\rVert_{L^p(I^d)} + \\lVert T_{N+1} f - T_{N+1} T^N f \\rVert_{L^p(I^d)} \\\\\n  &= \\lVert f -  T_{N+1} f \\rVert_{L^p(I^d)} + \\lVert \\lVert T_{N+1} (f - T^N f) \\rVert_{L^p(I)} \\rVert_{L^p(I^{d-1})} \\\\\n  &\\leq \\varepsilon_{N+1} + \\lVert \\lVert T \\rVert \\lVert  f - T^N f \\rVert_{L^p(I)} \\rVert_{L^p(I^{d-1})} \\\\\n  &= \\varepsilon_{N+1} + \\lVert T \\rVert \\lVert  f - T^N f \\rVert_{L^p(I^d)} \\\\ \n  &\\leq \\varepsilon_{N+1} + \\lVert T \\rVert \\sum_{i=1}^N  \\varepsilon_i \\lVert T \\rVert^{N-i} = \\sum_{i=1}^{N+1}  \\varepsilon_i \\lVert T \\rVert^{N+1-i}\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b\u306e\u3067$N+1$\u3067\u3082\u6210\u7acb\u3059\u308b. \n\\end{proof}\n\n\\begin{prop}\n  $I= (-\\pi,\\pi)$\u3068\u304a\u304d, $r,d \\geq 1$\u3068\u3059\u308b. $n = 1,2,3,\\ldots$\u306b\u5bfe\u3057\u3066\n  \\[\n  K_{n,r}^d := J_{\\left\\lfloor \\frac{n}{rd}  \\right\\rfloor+1,r}\n  \\]\n  \u3068\u304a\u304f. \u3053\u306e\u3068\u304d, \u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570$f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3067\u3042\u3063\u3066, \u3042\u308b$p \\in [1,\\infty]$\u306b\u3064\u3044\u3066$f \\in L^p(I)$\u3067\u3042\u308b\u3082\u306e\u306b\u5bfe\u3057\u3066,  \n  \\[\n  S_{n,r}^d (f)(x) := \\int_{-\\pi}^{\\pi} \\left( \\Delta_t^r(f)(x) + f(x) \\right) K_{n,r}^d(t) dt\n  \\]\n  \u3068\u5b9a\u3081\u308b\u3068, $S_{n,r}^d (f)$\u306f\u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570\u3067\u3042\u3063\u3066$L^p(I)$\u306b\u5c5e\u3057, \n  \\[\n  \\lVert S_{n,r}^d(f) \\rVert_{L^p(I)} \\leq (2^r +1) \\lVert f \\rVert_{L^p(I)} ~~~~(n=1,2,3,\\ldots).\n  \\]\n\\end{prop}\n\\begin{proof}\n  \u88dc\u984c\\ref{DefApproOperater}\u3068\u540c\u69d8\u306b\u3057\u3066$S_{n,r}^d(f)$\u306f\u4e09\u89d2\u591a\u9805\u5f0f\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u306e\u3067\u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570\u3067\u3042\u308b. \u4e0d\u7b49\u5f0f\u3092\u793a\u305d\u3046. \n  \u307e\u305a$p=\\infty$\u306e\u5834\u5408\u3092\u8003\u3048\u308b. \u3053\u306e\u3068\u304d, \u4efb\u610f\u306e$x \\in [-\\pi,\\pi]$\u306b\u5bfe\u3057\u3066\n  \\[\n  \\begin{aligned}\n  |S_{n,r}^d(f)(x)| \n  &\\leq \\int_{-\\pi}^\\pi |\\Delta_t^r(f)(x)+f(x)| K_{n,r}^d(t) dt \\\\\n  &\\leq \\left( \\sum_{k=0}^r \\binom{r}{k} + 1 \\right)\\lVert f \\rVert_{L^\\infty(I)} \\int_{-\\pi}^\\pi K_{n,r}^d(t) dt =\\left( 2^r + 1 \\right)\\lVert f \\rVert_{L^\\infty(I)}\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b\u306e\u3067\u3088\u3044. \u6b21\u306b$p < \\infty$\u306e\u5834\u5408\u3092\u8003\u3048\u308b. \u3053\u306e\u3068\u304d, \u7a4d\u5206\u5f62\u306eMinkowski\u306e\u4e0d\u7b49\u5f0f\u3088\u308a\n  \\[\n  \\begin{aligned}\n  \\lVert S_{n,r}^d(f) \\rVert_{L^p(I)} \n  &\\leq \\int_{-\\pi}^\\pi K_{n,r}^d(t) \\left( \\int_{-\\pi}^\\pi |\\Delta_t^r(f)(x)+f(x)|^p dx \\right)^{1\/p} dt \\\\\n  &\\leq \\int_{-\\pi}^\\pi K_{n,r}^d(t) \\left( \\sum_{k=0}^r \\binom{r}{k} + 1 \\right) \\lVert f \\rVert_{L^p(I)} dt = \\left( 2^r + 1 \\right)\\lVert f \\rVert_{L^p(I)}\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \n\\end{proof}\n\n\\begin{prop}\\label{JacksonEstimate1}\n  $I = (-\\pi,\\pi)$\u3068\u304a\u304d, $1 \\leq p \\leq \\infty,~r \\geq 2$\u3068\u3059\u308b. $f:\\mathbb{R}^d \\rightarrow \\mathbb{R}$\u306f\u5404\u5909\u6570\u3054\u3068\u306b\u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570\u3067$f \\in L^p(I)$\u3067\u3042\u308b\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $x = (x_1,\\ldots,x_d)$\u306b\u5bfe\u3057\u3066\n  \\[\n  S_{n,r,j}^d(f)(x) := \\int_{-\\pi}^\\pi \\left[ \\sum_{k=0}^r \\binom{r}{k} (-1)^{r-k} f(x_1,\\ldots,x_j+kt,\\ldots,x_d) + f(x) \\right] K_{n,r}^d(t) dt\n  \\]\n  \u3068\u304a\u304d, $k=1,\\ldots,d$\u306b\u5bfe\u3057\u3066$S^k = S_{n,r,k}^d S_{n,r,k-1}^d \\cdots S_{n,r,1}^d$\u3068\u304a\u304f\u3068, $S^d f$\u306f\u9ad8\u3005$n$\u6b21\u306e$d$\u5909\u6570\u4e09\u89d2\u591a\u9805\u5f0f\u3067\u3042\u308a, \n  \\[\n  \\lVert f - S^d f \\rVert_{L^p(I^d)} \\leq \\sum_{j=1}^d \\lVert f - S_{n,r,j}^d f \\rVert_{L^p(I^d)} (2^r+1)^{d-j} .\n  \\]\n\\end{prop}\n\\begin{proof}\n  $k=1,\\ldots,d$\u306b\u5bfe\u3057$S^k f$\u306f\u5404\u5909\u6570\u3054\u3068\u306b$2\\pi$\u5468\u671f\u95a2\u6570\u3067\u3042\u308a, $x_{k+1},\\ldots,x_d$\u3092\u56fa\u5b9a\u3059\u308b\u3068\u304d$S^k f$\u306f$x_1,\\ldots,x_k$\u306e\u95a2\u6570\u3068\u3057\u3066\u9ad8\u3005$k \\lfloor n\/d \\rfloor$\u6b21\u306e$k$\u5909\u6570\u4e09\u89d2\u591a\u9805\u5f0f\u3067\u3042\u308b\u3053\u3068\u3092\u5e30\u7d0d\u6cd5\u3067\u793a\u3059. \n  \u307e\u305a$S^1 f$\u304c\u5404\u5909\u6570\u3054\u3068\u306b$2\\pi$\u5468\u671f\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u306f$f$\u306e\u5468\u671f\u6027\u3068$S_{n,r,1}$\u306e\u5b9a\u7fa9\u3088\u308a\u660e\u3089\u304b.  \u305d\u3057\u3066$x_2,\\ldots,x_d$\u3092\u56fa\u5b9a\u3059\u308b\u3068\u304d$f$\u306e\u5468\u671f\u6027\u3088\u308a\u88dc\u984c\\ref{DefApproOperater}\u3068\u540c\u69d8\u306b$S_{n,r,1}(f)(x_1,\\ldots,x_d)$\u306f$x_1$\u306e\u95a2\u6570\u3068\u3057\u3066\u9ad8\u3005$\\lfloor n\/d \\rfloor$\u6b21\u306e$1$\u5909\u6570\u4e09\u89d2\u591a\u9805\u5f0f\u3067\u3042\u308b. \u6b21\u306b$k$\u3067\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3059\u308b. $S^{k+1} f$\u306e\u5468\u671f\u6027\u306f$S^k f$\u306e\u5468\u671f\u6027\u3088\u308a\u5f93\u3046. $x_1,\\ldots,x_i,x_{k+2},\\ldots,x_d$\u3092\u56fa\u5b9a\u3059\u308b. $S^k f$\u306e\u5468\u671f\u6027\u3088\u308a\u88dc\u984c\\ref{DefApproOperater}\u3068\u540c\u69d8\u306b$S^{k+1} f$\u306f$x_{k+1}$\u306e\u95a2\u6570\u3068\u3057\u3066\n  \\[\n  \\cos(m x_{k+1}),~\\sin(m x_{k+1}) ~~~~(m=0,\\ldots,\\lfloor n\/d \\rfloor)\n  \\]\n  \u306e\u7dda\u5f62\u7d50\u5408\u3067\u8868\u3055\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u308b.  \u305d\u3057\u3066, \u305d\u306e\u4fc2\u6570\u306e\u3046\u3061$x_1,\\ldots,x_k$\u306b\u4f9d\u5b58\u3059\u308b\u90e8\u5206\u306f\u88dc\u984c\\ref{DefApproOperater}\u306e\u8a3c\u660e\u304b\u3089\n  \\[\n  \\begin{aligned}\n  &\\int_{-m\\pi}^{m\\pi} \\frac{1}{k} (S^k f)(x_1,\\ldots,x_k,u,x_{k+2},\\ldots,x_d)\\cos(ju) du ~~~~(j=0,\\ldots,\\lfloor n\/d \\rfloor), \\\\\n  &\\int_{-m\\pi}^{m\\pi} \\frac{1}{k} (S^k f)(x_1,\\ldots,x_k,u,x_{k+2},\\ldots,x_d)\\cos(ju) du ~~~~(j=0,\\ldots,\\lfloor n\/d \\rfloor)\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u4e00\u65b9, \u5e30\u7d0d\u6cd5\u306e\u4eee\u5b9a\u3088\u308a$S^k f$\u306f$x_1,\\ldots,x_k$\u306e\u95a2\u6570\u3068\u3057\u3066\u9ad8\u3005$k \\lfloor n\/d \\rfloor$\u6b21\u306e$k$\u5909\u6570\u4e09\u89d2\u591a\u9805\u5f0f\u3067\u3042\u308b\u306e\u3067\n  \\[\n  (S^k f)(x_1,\\ldots,x_k,u,x_{k+2},\\ldots,x_d) = \\sum_{|\\alpha|\\leq k \\lfloor n\/d \\rfloor} c_{u,x_{k+2},\\ldots,x_d}^{(\\alpha)} \\exp(i(\\alpha_1 x_1 + \\cdots + \\alpha_i x_k))\n  \\]\n  \u3068\u8868\u3055\u308c\u308b. \u3088\u3063\u3066, \n  \\[\n  \\cos(mx) = \\frac{\\exp(imx) + \\exp(-imx)}{2}, ~~ \\sin(mx) = \\frac{\\exp(imx) - \\exp(-imx)}{2}\n  \\]\n  \u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308c\u3070$S^{k+1} f$\u306f$x_1,\\ldots,x_{k+1}$\u306e\u95a2\u6570\u3068\u3057\u3066\n  \\[\n  \\exp(i (\\alpha_1 x_1 + \\alpha_{k+1} x_{k+1})) ~~~~(|\\alpha| \\leq (k+1) \\lfloor n\/d \\rfloor)\n  \\]\n  \u306e\u7dda\u5f62\u7d50\u5408\u3067\u8868\u3055\u308c\u308b. \u3053\u308c\u3067$k+1$\u306e\u6210\u7acb\u304c\u793a\u3055\u308c\u305f. \u3053\u306e\u3053\u3068\u3088\u308a\u524d\u534a\u306e\u4e3b\u5f35\u306f\u6210\u7acb\u3059\u308b.  \u5f8c\u534a\u306e\u4e3b\u5f35\u306b\u3064\u3044\u3066\u306f\u524d\u547d\u984c\u3068\u88dc\u984c\\ref{LemforMultiJacksonEst}\u3088\u308a\u5f93\u3046. \n\\end{proof}\n\n\\subsubsection{Sobolev\u7a7a\u9593\u306e\u5143\u306e\u5834\u5408}\n\u672c\u5c0f\u7bc0\u3067\u306f, $r \\geq 1$, $1 \\leq p \\leq \\infty$\u306b\u5bfe\u3057\u3066\u5b9a\u6570$C_{r,p} > 0$\u304c\u5b58\u5728\u3057\u3066, \u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570$f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3067\u4efb\u610f\u306e\u6709\u754c\u958b\u533a\u9593$J \\subset \\mathbb{R}$\u306b\u3064\u3044\u3066$f \\in W^{r,p}(J)$\u3067\u3042\u308b\u3082\u306e\u306b\u5bfe\u3057\u3066\n\\[\n\\omega_r(f)_{p,I}\\left(\\frac{1}{n}\\right) \\leq C_{r,p} \\frac{1}{n^r} \\lVert f^{(r)} \\rVert_{L^p(I)} ~~~~(n = 1,2,3,\\ldots) ~~~~\\tag{1}\n\\]\n\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u304f(\u305f\u3060\u3057, $I=(-\\pi,\\pi)$\u3067\u3042\u308b). \u3053\u308c\u304c\u793a\u3055\u308c\u308c\u3070\u6b21\u306e\u3053\u3068\u304c\u308f\u304b\u308b. \n\\begin{thm}\\label{JacksonEstimate3}\n  $I = (-\\pi,\\pi)$\u3068\u304a\u304d, $1 \\leq d < \\infty$, $1 \\leq p \\leq \\infty$, $r \\geq 2$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u6b21\u3092\u307f\u305f\u3059\u5b9a\u6570$C$\u304c\u5b58\u5728\u3059\u308b. \n  $f:\\mathbb{R}^d \\rightarrow \\mathbb{R}$\u304c\u5404\u5909\u6570\u3054\u3068\u306b\u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570\u304b\u3064\u4efb\u610f\u306e\u6709\u754c\u958b\u533a\u9593$J \\subset \\mathbb{R}$\u306b\u3064\u3044\u3066$f \\in W^{r,p}(J)$\u3067\u3042\u308b\u306a\u3089\u3070, \n  \\[\n  \\begin{aligned}\n  E_{n,p}(f) &:= \\inf \\{ \\lVert T - f \\rVert_{L^p(I^d)} \\mid T\\mbox{\u306f\u9ad8\u3005}n\\mbox{\u6b21\u306e}d\\mbox{\u5909\u6570\u4e09\u89d2\u591a\u9805\u5f0f} \\} \\\\\n  &~\\leq C \\frac{1}{n^r} \\sum_{j=1}^d \\left\\lVert \\partial_j^r f \\right\\rVert_{L^p(I^d)}  ~~~~(n=1,2,3,\\ldots).\n  \\end{aligned}\n  \\]\n\\end{thm}\n\\begin{proof}\n  \u547d\u984c\\ref{JacksonEstimate1}\u3088\u308a$S^d f$\u306f\u9ad8\u3005$n$\u6b21\u306e$d$\u5909\u6570\u4e09\u89d2\u591a\u9805\u5f0f\u3067\u3042\u308a, \n  \\[\n  \\lVert f - S^d f \\rVert_{L^p(I^d)} \\leq \\sum_{j=1}^d \\lVert f - S_{n,r,j}^d f \\rVert_{L^p(I^d)} (2^r+1)^{d-j} .\n  \\]\n  \u3068\u306a\u308b. $f,S_{n,r,j}^d f$\u3092$x_j$\u306e\u95a2\u6570\u3068\u307f\u306a\u3059\u3068\u304d\u5b9a\u7406\\ref{JacksonEstimate0}\u3068\u540c\u69d8\u306b\u3057\u3066\u5b9a\u6570$C_j$\u304c\u5b58\u5728\u3057\u3066\n  \\[\n  \\lVert f - S_{n,r,j}^d \\rVert_{L^p(I)} \\leq C \\omega_r(f)_{p,I}\\left( \\frac{1}{n} \\right) ~~~~(n=1,2,3,\\ldots)\n  \\]\n  \u3068\u306a\u308b. \u3057\u305f\u304c\u3063\u3066, (1)\u3088\u308a\n  \\[\n  \\begin{aligned}\n  \\lVert f - S^d f \\rVert_{L^p(I^d)} \n  &\\leq \\sum_{j=1}^d \\lVert f - S_{n,r,j}^d f \\rVert_{L^p(I^d)} (2^r+1)^{d-j} \\\\\n  &\\leq (2^r+1)^d \\sum_{j=1}^d \\lVert \\lVert f - S_{n,r,j}^d f \\rVert_{L^p(I)} \\rVert_{L^p(I^{d-1})} \\\\\n  &\\leq (2^r+1)^d C \\sum_{j=1}^d \\left\\lVert \\omega_r(f)_{p,I}\\left( \\frac{1}{n} \\right) \\right\\rVert_{L^p(I^{d-1})} \\\\\n  &\\leq C' \\frac{1}{n^r} \\sum_{j=1}^d \\left\\lVert \\left\\lVert \\partial_j^r f \\right\\rVert_{L^p(I)} \\right\\rVert_{L^p(I^{d-1})} = C' \\frac{1}{n^r} \\sum_{j=1}^d \\left\\lVert \\partial_j^r f \\right\\rVert_{L^p(I^d)}\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3053\u308c\u3067\u793a\u305b\u305f. \n\\end{proof}\n\u3055\u3089\u306b\u3053\u306e\u3053\u3068\u304b\u3089\u6b21\u306e\u3053\u3068\u3082\u308f\u304b\u308b. \n\\begin{thm}\\label{JacksonEstimate4}\n  $1 \\leq d < \\infty$, $1 \\leq p \\leq \\infty$, $r \\geq 2$\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, $n$\u306b\u4f9d\u5b58\u3057\u306a\u3044\u5b9a\u6570$C$\u304c\u5b58\u5728\u3057\u3066, \u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570$f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3067\u4efb\u610f\u306e\u6709\u754c\u958b\u533a\u9593$J \\subset \\mathbb{R}$\u306b\u5bfe\u3057\u3066$f \\in W^{r,p}(J)$\u3067\u3042\u308b\u3082\u306e\u306b\u5bfe\u3057\u3066\n  \\[\n  \\begin{aligned}\n  E_{n,p}(f) &:= \\inf \\{ \\lVert P - f \\rVert_{L^p((-1,1)^d)} \\mid T\\mbox{\u306f\u9ad8\u3005}n\\mbox{\u6b21\u306e}d\\mbox{\u5909\u6570\u591a\u9805\u5f0f} \\} \\\\\n  &~\\leq C \\frac{1}{n^r}  \\left\\lVert  f \\right\\rVert_{W^{r,p}((-1,1)^d)}  ~~~~(n=1,2,3,\\ldots).\n  \\end{aligned}\n  \\]\n\\end{thm}\n\\begin{proof}\n  \u6b21\u3092\u307f\u305f\u3059\u6709\u754c\u7dda\u5f62\u4f5c\u7528\u7d20\u304c\u5b58\u5728\u3059\u308b(\u62e1\u5f35\u4f5c\u7528\u7d20. \u5bae\u5cf6\\cite{MiyajimaSobolev}\u3092\u53c2\u7167). \n  \\[\n  T:W^{r,p}((-1,1)^d) \\rightarrow W^{r,p}(\\mathbb{R}^d),~~ (Tf)(x) = f(x) ~~~(\\forall x \\in (-1,1)^d).\n  \\]\n  $0 \\leq \\varphi \\in C_0^\\infty(\\mathbb{R}^d)$\u3092$(-1,1)^d$\u4e0a\u3067$1$\u3092\u3068\u308a$(-3\/2,3\/2)^d$\u306e\u5916\u3067$0$\u3092\u3068\u308b\u3082\u306e\u3068\u3059\u308b. $x \\in \\mathbb{R}^d$\u306b\u5bfe\u3057\u3066$g(x) := \\varphi(x)(Tf)(x)$\u3068\u304a\u304f. \u305d\u3057\u3066\n  $x = (x_1,\\ldots,x_d) \\in \\mathbb{R}^d$\u306b\u5bfe\u3057\u3066\n  \\[\n  h(x) := g(2\\cos(x_1),\\ldots,2\\cos(x_d))\n  \\]\n  \u3068\u304a\u304f. \u3059\u308b\u3068, $(-3\/2,3\/2)^d$\u306e\u5916\u3067$g=0$\u3086\u3048, $h$\u306f\u5404\u5909\u6570\u3054\u3068\u306b$2\\pi$\u5468\u671f\u306e\u5468\u671f\u95a2\u6570\u304b\u3064\u4efb\u610f\u306e\u6709\u754c\u958b\u533a\u9593$J \\subset \\mathbb{R}$\u306b\u5bfe\u3057\u3066$h \\in W^{r,p}(J)$\u3067\u3042\u308b. \u3057\u305f\u304c\u3063\u3066, \u7279\u306b, \u547d\u984c\\ref{JacksonEstimate1}\u3088\u308a$S^d h$\u306f\u9ad8\u3005$n$\u6b21\u306e$d$\u5909\u6570\u4e09\u89d2\u591a\u9805\u5f0f\u3067\u3042\u308b. \u307e\u305f, $h$\u306f\u5404\u5909\u6570\u3054\u3068\u306b\u5076\u95a2\u6570\u3067\u3042\u308b\u306e\u3067\u547d\u984c\\ref{JacksonEstimate1}\u306e\u8a3c\u660e\u3088\u308a$S^d h$\u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b:\n  \\[\n  (S^d h)(x) = \\sum_{|\\alpha| \\leq n} c_{\\alpha} \\prod_{j=1}^d \\cos(\\alpha_j x_j) ~~~~(c_{\\alpha} \\in \\mathbb{R}).\n  \\]\n  \u3053\u3053\u3067\u6574\u6570$k \\geq 0$\u306b\u5bfe\u3057\u3066\u9ad8\u3005$k$\u6b21\u306e$1$\u5909\u6570\u591a\u9805\u5f0f$T_k$\u304c\u5b58\u5728\u3057\u3066\n  \\[\n  T_k(2\\cos x) = \\cos(kx) ~~~~(\\forall x \\in \\mathbb{R})\n  \\]\n  \u3068\u306a\u308b(Chebyshev\u591a\u9805\u5f0f). \u305d\u3053\u3067\u9ad8\u3005$n$\u6b21\u306e$d$\u5909\u6570\u591a\u9805\u5f0f$P_f$\u3092\n  \\[\n  P_f(t) = \\sum_{|\\alpha| \\leq n} c_\\alpha \\prod T_{\\alpha_j}(t_j)\n  \\]\n  \u3067\u5b9a\u3081, $\\Phi(x) = (2\\cos(x_1),\\ldots,2\\cos(x_d))$\u3068\u304a\u304f\u3068, $P(\\Phi(x)) = (S^d h)(x)$\u3068\u306a\u308b.  \u3088\u3063\u3066\n  , \u5909\u6570\u5909\u63db\u3068\u5b9a\u7406\\ref{JacksonEstimate3}\u306e\u8a3c\u660e\u3088\u308a\n  \\[\n  \\begin{aligned}\n  \\lVert f - P_f \\rVert_{L^p((-1,1)^d)} \n  &\\leq \\lVert g - P_f \\rVert_{L^p((-2,2)^d)} \\leq C \\lVert g \\circ \\Phi - P_f \\circ \\Phi \\rVert_{L^p((-\\pi,\\pi)^d)} \\\\\n  &= \\lVert h - S^d h \\rVert_{L^p((-\\pi,\\pi)^d)} \\leq \\frac{C'}{n^r} \\sum_{j=1}^d \\lVert \\partial_j^r h \\rVert_{L^p((-\\pi,\\pi)^d)} \\\\\n  &\\leq \\frac{C''}{n^r} \\lVert  g \\rVert_{W^{r,p}((-2,2)^d)} ~~~~(\\because g(x) = 0 ~(\\forall x \\notin (-3\/2,3\/2))) \\\\\n  &\\leq \\frac{C''}{n^r} \\lVert Tf \\rVert_{W^{r,p}((-2,2)^d)} \\leq \\frac{C''}{n^r} \\lVert T \\rVert \\lVert f \\rVert_{W^{r,p}((-1,1)^d)} \n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \n\\end{proof}\n\u3055\u3066, \u5c0f\u7bc0\u306e\u306f\u3058\u3081\u306b\u8ff0\u3079\u305f\u3053\u3068\u3092\u4ee5\u4e0b\u3067\u793a\u3057\u3066\u3044\u3053\u3046. \n\\begin{prop}\n  $\\mathbb{R}$\u4e0a\u306e\u95a2\u6570$M_r ~~(r=1,2,3,\\ldots)$\u3092\u6b21\u306e\u3088\u3046\u306b\u5e30\u7d0d\u7684\u306b\u5b9a\u3081\u308b. \n  \\[\n  M_1 := \\chi_{[0,1]}, ~~ M_r(x) := (M_{r-1}*M_1)(x) = \\int_{\\mathbb{R}} M_{r-1}(x-y)M_1(y) dy\n  \\]\n  \u3059\u308b\u3068, $\\mathrm{supp}M_r \\subset [0,r]$, $0 \\leq M_r \\leq 1$\u3068\u306a\u308b. \n\\end{prop}\n\\begin{proof}\n  $r=1$\u3067\u306e\u6210\u7acb\u306f\u660e\u3089\u304b. $r-1$\u3067\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b. \u3059\u308b\u3068, \n  \\[\n  0 \\leq M_r(x) = \\int_{\\mathbb{R}} M_{r-1}(x-y)M_1(y) dy \\leq \\int_{[0,1]} 1 dy = 1 ~~~~(\\forall x)\n  \\]\n  \u3067\u3042\u308a, $x \\in (-\\infty,0)\\cup (r,\\infty)$\u3068\u3059\u308b\u3068$y \\in [0,1]$\u306b\u5bfe\u3057\u3066$x-y \\notin [0,r-1]$\u3067\u3042\u308b\u306e\u3067$M_r(x)=0$\u3067\u3042\u308b. \n\\end{proof}\n\\begin{prop}\\label{IncDiffRepresent}\n  $I = (-\\pi,\\pi) \\subset \\mathbb{R}$\u3068\u304a\u304d, $r = 1,2,3,\\ldots$\u306b\u5bfe\u3057\u3066, \n  \\[\n  M_r(x,h) := \\frac{1}{h}M_r\\left( \\frac{x}{h} \\right) ~~~~(x \\in \\mathbb{R},~ h > 0)\n  \\]\n  \u3068\u5b9a\u3081\u308b. \u3053\u306e\u3068\u304d, \u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570$f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3067\u4efb\u610f\u306e\u6709\u754c\u958b\u533a\u9593$J \\subset \\mathbb{R}$\u306b\u5bfe\u3057\u3066$f \\in W^{r,p}(J)$\u3067\u3042\u308b\u3082\u306e\u3068$x \\in I$\u306b\u3064\u3044\u3066\n  \\[\n  h^{-r} \\Delta_h^r(f)(x) = h^{-r} \\sum_{k=0}^r \\binom{r}{k} (-1)^{r-k} f(x+kh) = \\int_{\\mathbb{R}} f^{(r)}(x+t)M_r(t,h) dt\n  \\]\n  \u304c\u6210\u308a\u7acb\u3064. \n\\end{prop}\n\\begin{proof}\n  $r$\u306b\u95a2\u3059\u308b\u5e30\u7d0d\u6cd5\u306b\u3088\u308b. \u307e\u305a$r=1$\u306e\u3068\u304d\u306f, \u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570$f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3067\u4efb\u610f\u306e\u6709\u754c\u958b\u533a\u9593$J \\subset \\mathbb{R}$\u306b\u3064\u3044\u3066$f \\in W^{r,p}(J)$\u3067\u3042\u308b\u3082\u306e\u306b\u5bfe\u3057\u3066\n  \\[\n  \\begin{aligned}\n  \\int_{\\mathbb{R}} f^{(1)}(x+t) M_1(t,h) dt \n  &= \\int_{\\mathbb{R}} f^{(1)}(x+t) h^{-1} \\chi_[0,1](h^{-1}t) dt \\\\\n  &= \\int_{[0,h]} f^{(1)}(x+t) dt = f(x+h)-f(x)\n  \\end{aligned}\n  \\]\n  \u3060\u304b\u3089\u6210\u7acb\u3059\u308b. \u6b21\u306b$r-1$\u3067\u6210\u7acb\u3059\u308b\u3068\u3059\u308b. \u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570$f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3067\u4efb\u610f\u306e\u6709\u754c\u958b\u533a\u9593$J \\subset \\mathbb{R}$\u306b\u3064\u3044\u3066$f \\in W^{r,p}(J)$\u3067\u3042\u308b\u3082\u306e\u3092\u4efb\u610f\u306b\u53d6\u308b. \u3053\u306e\u3068\u304d$\\Delta_h^1(f)$\u306f\u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570\u3067\u4efb\u610f\u306e\u6709\u754c\u958b\u533a\u9593$J \\subset \\mathbb{R}$\u306b\u3064\u3044\u3066$\\Delta_h^1(f) \\in W^{r,p}(J)$\u3067\u3042\u308b\u306e\u3067\u5e30\u7d0d\u6cd5\u306e\u4eee\u5b9a\u3088\u308a\n  \\[\n  \\begin{aligned}\n  h^{-r} \\Delta_h^r(f,x) \n  &= h^{-1} h^{-(r-1)} \\Delta_h^{r-1}(\\Delta_h^1(f))(x) \\\\\n  &= h^{-1} \\int_{\\mathbb{R}} (\\Delta_h^1(f))^{(r-1)}(x+t) M_{r-1}(t,h) dt \\\\\n  &= h^{-1} \\int_{\\mathbb{R}} \\left( f^{(r-1)}(x+h+t) - f^{(r-1)}(x+t) \\right) M_{r-1}(t,h) dt \\\\\n  &= h^{-1} \\int_{\\mathbb{R}} \\left( \\int_0^h f^{(r)}(x+v+t) dv \\right) M_{r-1}(t,h) dt \\\\\n  &= h^{-1} \\int_{\\mathbb{R}} \\left( \\int_0^h f^{(r)}(x+v+uh) dv \\right) M_{r-1}(u) du \\\\\n  &= \\int_{\\mathbb{R}} \\left( \\int_0^1 f^{(r)}(x+(v+u)h) dv \\right) M_{r-1}(u) du \\\\\n  &= \\int_0^1 \\left( \\int_{\\mathbb{R}} f^{(r)}(x+(v+u)h) M_{r-1}(u) du \\right) dv \\\\\n  &= \\int_0^1 \\left( \\int_{\\mathbb{R}} f^{(r)}(x+wh) M_{r-1}(w-v) dw \\right) dv \\\\\n  \\end{aligned}\n  \\]\n  \\[\n  \\begin{aligned}\n  &= \\int_{\\mathbb{R}} \\left( f^{(r)}(x+wh) \\int_0^1 M_{r-1}(w-v) dv \\right) dw \\\\\n  &= \\int_{\\mathbb{R}} f^{(r)}(x+wh) M_r(w) dw \\\\\n  &= \\int_{\\mathbb{R}} f^{(r)}(x+t) M_r\\left( \\frac{t}{h}\\right) \\frac{1}{h} dt \\\\\n  &= \\int_{\\mathbb{R}} f^{(r)} (x+t) M_r(t,h) dt.\n  \\end{aligned}\n  \\]\n\\end{proof}\n\\begin{rem}\n  \u4e0a\u306e\u547d\u984c\u306e\u8a3c\u660e\u3067Sobolev\u7a7a\u9593\u306e\u5143\u306b\u5bfe\u3059\u308b\u5fae\u5206\u7a4d\u5206\u5b66\u306e\u57fa\u672c\u5b9a\u7406\u3092\u7528\u3044\u305f. \u3053\u306e\u8a3c\u660e\u306f\u4f8b\u3048\u3070\u5bae\u5cf6\\cite{MiyajimaSobolev}\u306e\u547d\u984c3.4\u3092\u53c2\u7167\u3055\u308c\u305f\u3044. \n\\end{rem}\n\\begin{thm}\n  $I = (-\\pi,\\pi) \\subset \\mathbb{R}$\u3068\u304a\u304f. $r \\geq 1$, $1 \\leq p \\leq \\infty$\u306b\u5bfe\u3057\u3066\u5b9a\u6570$C_{r,p} > 0$\u304c\u5b58\u5728\u3057\u3066, \u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570$f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3067\u4efb\u610f\u306e\u6709\u754c\u958b\u533a\u9593$J \\subset \\mathbb{R}$\u306b\u3064\u3044\u3066$f \\in W^{r,p}(J)$\u3067\u3042\u308b\u3082\u306e\u306b\u5bfe\u3057\u3066\n  \\[\n  \\omega_{r}(f)_{p,I}(t) = \\sup_{0 < h \\leq t} \\lVert \\Delta_h^r(f) \\rVert_{L^p(I)} \\leq C_{r,p} t^r \\lVert f^{(r)} \\rVert_{L^p(I)} ~~~~(\\forall t \\in (0,2\\pi\/r)).\n  \\]\n\\end{thm}\n\\begin{proof}\n  \u547d\u984c\\ref{IncDiffRepresent}\u3088\u308a$x \\in I$\u306b\u5bfe\u3057\u3066\n  \\[\n  \\Delta_h^r(f)(x) = h^r \\int_{\\mathbb{R}} f^{(r)}(x+t)M_r(t,h) dt\n  \\]\n  \u3067\u3042\u308b\u306e\u3067, $r,p$\u306e\u307f\u306b\u4f9d\u5b58\u3059\u308b\u5b9a\u6570$C_{r,p} > 0$\u304c\u5b58\u5728\u3057\u3066\n  \\[\n  \\left\\lVert \\int_{\\mathbb{R}} f^{(r)}(\\cdot+t)M_r(t,h) dt \\right\\rVert_{L^p(I)} \\leq C_{r,p} \\lVert f^{(r)} \\rVert_{L^p(I)} ~~~~(\\forall h \\in (0,2\\pi\/r) )\n  \\]\n  \u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044. \u307e\u305a$p=\\infty$\u306e\u5834\u5408\u306f\n  \\[\n  \\begin{aligned}\n  \\int_{\\mathbb{R}} f^{(r)} (x+t) M_r(t,h) dt \n  &= \\int_{\\mathbb{R}} f^{(r)}(x+t) M_r\\left( \\frac{t}{h}\\right) \\frac{1}{h} dt  \\\\\n  &= \\int_{\\mathbb{R}} f^{(r)}(x+wh) M_r(w) dw.\n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b\u3053\u3068\u3068$\\mathrm{supp}M_r \\subset [0,r]$\u3088\u308a\n  \\[\n  \\left\\lVert \\int_{\\mathbb{R}} f^{(r)}(\\cdot+t)M_r(t,h) dt \\right\\rVert_{L^\\infty(I)} \\leq r \\lVert f^{(r)} \\rVert_{L^\\infty(I)}\n  \\]\n  \u3068\u306a\u308a\u6210\u7acb\u3059\u308b. \u6b21\u306b$p < \\infty$\u306e\u5834\u5408\u3092\u8003\u3048\u308b. Minkowski\u306e\u4e0d\u7b49\u5f0f(\u7cfb\\ref{CorOfMinkowskiInequality})\u3088\u308a\n  \\[\n  \\begin{aligned}\n  &\\left( \\int_I \\left\\lvert \\int_{\\mathbb{R}} f^{(r)}(x+t)M_r(t,h) dt  \\right\\rvert^p dx \\right)^{1\/p} \\\\\n  &\\leq \\left( \\int_I \\left( \\int_{\\mathbb{R}} \\lvert f^{(r)}(t)\\rvert M_r(t-x,h) dt  \\right)^p dx \\right)^{1\/p} \\\\\n  &\\leq \\int_{\\mathbb{R}} |f^{(r)}(t)| \\left( \\int_I M_r(t-x,h)^p dx \\right)^{1\/p} dt \n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308b. \u3053\u3053\u3067, \n  \\[\n  \\begin{aligned}\n  \\int_I M_r(t-x,h)^p dx\n  &= \\int_{-\\pi}^{\\pi} \\frac{1}{h}M_r\\left(\\frac{t-x}{h} \\right)^p dx \\\\\n  &= \\int_{\\frac{t-\\pi}{h}}^{\\frac{t+\\pi}{h}} M_r(y)^p dy \n  \\end{aligned}\n  \\]\n  \u3067\u3042\u308a$\\mathrm{supp}M_r \\subset [0,r]$\u3067\u3042\u308b\u306e\u3067$t \\in (-\\infty,-\\pi] \\cup [\\pi+rh,\\infty)$\u306e\u3068\u304d\n  \\[\n  \\int_I M_r(t-x,h)^p dx = 0\n  \\]\n  \u3067\u3042\u308b. \u3088\u3063\u3066, $h \\in (0,2\\pi\/r)$\u306b\u6ce8\u610f\u3059\u308c\u3070, \n   \\[\n  \\begin{aligned}\n  &\\int_{\\mathbb{R}} |f^{(r)}(t)| \\left( \\int_I M_r(t-x,h)^p dx \\right)^{1\/p} dt \\\\\n  &= \\int_{-\\pi}^{3\\pi} |f^{(r)}(t)| \\left( \\int_{\\frac{t-\\pi}{h}}^{\\frac{t+\\pi}{h}} M_r(y)^p dy  \\right)^{1\/p} dt \\\\\n  &\\leq \\int_{-\\pi}^{3\\pi} |f^{(r)}(t)| \\left( \\int_0^r 1 dy  \\right)^{1\/p} dt = 2r^{1\/p} \\int_{-\\pi}^{\\pi} |f^{(r)}(t)| dt \n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3053\u308c\u3067$p=1$\u306e\u5834\u5408\u306e\u6210\u7acb\u306f\u308f\u304b\u308b. $1 < p < \\infty$\u306e\u5834\u5408\u306f$1\/p+1\/q=1$\u3068\u3059\u308b\u3068H\\\"{o}lder\u306e\u4e0d\u7b49\u5f0f\u3088\u308a\n  \\[\n  \\int_{-\\pi}^{\\pi} |f^{(r)}(t)| dt \\leq \\left(\\int_{-\\pi}^\\pi 1^q dt \\right)^{q}  \\left(\\int_{-\\pi}^\\pi |f^{(r)}(t)|^p dt \\right)^{p} \n  \\]\n  \u3060\u304b\u3089\u6210\u7acb\u3059\u308b. \n\\end{proof}\n\\begin{cor}\n  $I = (-\\pi,\\pi) \\subset \\mathbb{R}$\u3068\u304a\u304f. $r \\geq 1$, $1 \\leq p \\leq \\infty$\u306b\u5bfe\u3057\u3066\u5b9a\u6570$C_{r,p} > 0$\u304c\u5b58\u5728\u3057\u3066, \u5468\u671f$2\\pi$\u306e\u5468\u671f\u95a2\u6570$f:\\mathbb{R} \\rightarrow \\mathbb{R}$\u3067\u4efb\u610f\u306e\u6709\u754c\u958b\u533a\u9593$J \\subset \\mathbb{R}$\u306b\u3064\u3044\u3066$f \\in W^{r,p}(J)$\u3067\u3042\u308b\u3082\u306e\u306b\u5bfe\u3057\u3066\n  \\[\n  \\omega_{r}(f)_{p,I}\\left(\\frac{1}{n}\\right) \\leq \\frac{C_{r,p}}{n^r} \\lVert f^{(r)} \\rVert_{L^p(I)} ~~~~(n=1,2,3,\\ldots).\n  \\]\n\\end{cor}\n\\begin{proof}\n  $N_{r,p} \\in \\mathbb{N}$\u3092\u5341\u5206\u5927\u304d\u304f\u53d6\u308c\u3070$n \\geq N_{r,p}$\u306b\u5bfe\u3057\u3066$1\/n \\in (0,2\\pi\/r)$\u3068\u306a\u308b. \u3088\u3063\u3066, \u3059\u3050\u4e0a\u306e\u5b9a\u7406\u3088\u308a\n  \\[\n  \\omega_{r}(f)_{p,I}\\left(\\frac{1}{n}\\right) \\leq \\frac{C_{r,p}}{n^r} \\lVert f^{(r)} \\rVert_{L^p(I)} ~~~~(\\forall n \\geq N_{r,p}).\n  \\]\n  \u3068\u306a\u308b. \u305d\u3053\u3067$C_{r,p}$\u3092\u5fc5\u8981\u306b\u5fdc\u3058\u3066\u5927\u304d\u304f\u3059\u308c\u3070$n=1,\\ldots,N_{r,p}$\u306b\u3064\u3044\u3066\u3082\n  \\[\n  \\omega_{r}(f)_{p,I}\\left(\\frac{1}{n}\\right) \\leq \\frac{C_{r,p}}{n^r} \\lVert f^{(r)} \\rVert_{L^p(I)}\n  \\]\n  \u3068\u306a\u308b. \u3053\u308c\u3067\u793a\u305b\u305f. \n\\end{proof}\n\n\\subsection{Baire\u306e\u30ab\u30c6\u30b4\u30ea\u30fc\u5b9a\u7406}\n\u672c\u7bc0\u3067\u306f\u5b8c\u5099\u8ddd\u96e2\u7a7a\u9593\u306b\u304a\u3051\u308bBaire\u306e\u30ab\u30c6\u30b4\u30ea\u30fc\u5b9a\u7406\u306b\u3064\u3044\u3066\u8ff0\u3079\u308b. \n\\begin{thm}[Baire\u306e\u30ab\u30c6\u30b4\u30ea\u30fc\u5b9a\u7406] \\\\\n  $(X,d)$\u3092\u5b8c\u5099\u8ddd\u96e2\u7a7a\u9593\u3068\u3057, $(V_n)_{n\\in \\mathbb{N}}$\u3092$X$\u306b\u304a\u3044\u3066\u7a20\u5bc6\u306a\u958b\u96c6\u5408\u304b\u3089\u306a\u308b\u5217\u3068\u3059\u308b. \u3053\u306e\u3068\u304d $\\bigcap_{n\\in\\mathbb{N}}V_n$\u306f$X$\u306b\u304a\u3044\u3066\u7a20\u5bc6\u3067\u3042\u308b. \n\\end{thm}\n\\begin{proof}\n  \u4efb\u610f\u306e$x_0\\in X$\u3068$r_0\\in (0,\\infty)$\u306b\u5bfe\u3057, \n  \\[\n  B(x_0,r_0)\\cap \\bigcap_{n\\in \\mathbb{N}}V_n\\neq\\emptyset\\quad\\quad(*)\n  \\]\n  \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044. $x_0\\in X=\\overline{V_1}$\u3088\u308a$B(x_0,r_0)\\cap V_1\\neq\\emptyset$\u3067\u3042\u308b. \u3088\u3063\u3066, \n  \\[\n  \\overline{B}(x_1,r_1)\\subset B(x_0,r_0)\\cap V_1,\\quad 0<r_1\\leq\\frac{r_0}{2}\n  \\]\n  \u306a\u308b\u9589\u7403$\\overline{B}(x_1,r_1)=\\{x\\in X:d(x,x_1)\\leq r_1\\}$\u304c\u53d6\u308c\u308b. $x_1\\in X=\\overline{V_2}$\u3088\u308a$B(x_1,r_1)\\cap V_2\\neq\\emptyset$ \u3067\u3042\u308b\u304b\u3089,  \n  \\[\n  \\overline{B}(x_2,r_2)\\subset B(x_1,r_1)\\cap V_2,\\quad 0<r_2\\leq\\frac{r_0}{2^2}\n  \\]\n  \u306a\u308b\u9589\u7403$\\overline{B}(x_2,r_2)$\u304c\u53d6\u308c\u308b. \u540c\u69d8\u306e\u3053\u3068\u3092\u7e70\u308a\u8fd4\u305b\u3070\u9589\u7403\u306e\u5217 $(\\overline{B}(x_n,r_n))_{n\\in \\mathbb{N}}$\u3067, \n  \\[\n  \\overline{B}(x_{n},r_{n})\\subset B(x_{n-1},r_{n-1})\\cap V_{n},\\quad 0<r_n\\leq\\frac{r_0}{2^n}\\quad(\\forall n\\in \\mathbb{N})\\quad\\quad(**)\n  \\]\n  \u3092\u6e80\u305f\u3059\u3082\u306e\u304c\u53d6\u308c\u308b. \u3053\u306e\u3068\u304d, \n  \\[\n  \\bigcap_{n\\in \\mathbb{N}}\\overline{B}(x_n,r_n)\n  \\subset \\bigcap_{n \\in \\mathbb{N}} B(x_{n-1},r_{n-1}) \\cap \\bigcap_{n\\in \\mathbb{N}}V_n\n  \\subset B(x_0,r_0)\\cap \\bigcap_{n\\in \\mathbb{N}}V_n\n  \\]\n  \u3067\u3042\u308b. \u3088\u3063\u3066$(*)$\u3092\u793a\u3059\u305f\u3081\u306b\u306f$\\bigcap_{n\\in \\mathbb{N}}\\overline{B}(x_n,r_n)\\neq\\emptyset$\u3092\u793a\u305b\u3070\u3088\u3044. $m>n$ \u306a\u308b\u4efb\u610f\u306e$n,m\\in \\mathbb{N}$\u306b\u5bfe\u3057\u3066$(**)$\u3088\u308a, \n  \\[\n  d(x_m,x_n)\\leq d(x_m,x_{m-1})+\\ldots+d(x_{n+1},x_n)\\leq r_{m-1}+\\ldots+r_n\\leq2\\frac{r_0}{2^n}\n  \\]\n  \u3067\u3042\u308b. \u3053\u308c\u3088\u308a$(x_n)_{n\\in \\mathbb{N}}$\u306fCauchy\u5217\u3067\u3042\u308b\u306e\u3067, $X$\u306e\u5b8c\u5099\u6027\u3088\u308a\u3042\u308b$x\\in X$\u306b\u53ce\u675f\u3059\u308b. $x\\in \\bigcap_{n\\in \\mathbb{N}}\\overline{B}(x_n,r_n)$\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305d\u3046. \u4efb\u610f\u306e$n\\in \\mathbb{N}$\u3068$\\epsilon\\in (0,\\infty)$\u306b\u5bfe\u3057\u3066, $m\\geq n$\u304b\u3064 $d(x_m,x)<\\epsilon$\u3092\u6e80\u305f\u3059$m\\in \\mathbb{N}$\u3092\u53d6\u308c\u3070, $x_m\\in B(x,\\epsilon)\\cap \\overline{B}(x_n,r_n)\\neq\\emptyset$\u3067\u3042\u308b.\n  \u3088\u3063\u3066$\\epsilon$\u306e\u4efb\u610f\u6027\u3088\u308a$x\\in \\overline{B}(x_n,r_n)$\u3067\u3042\u308a, $n\\in \\mathbb{N}$\u306e\u4efb\u610f\u6027\u3088\u308a$x\\in \\bigcap_{n\\in \\mathbb{N}}\\overline{B}(x_n,r_n)$\u3067\u3042\u308b. \n\\end{proof}\n\\begin{cor}\\label{BaireCategoryTheorem}\n  $X$\u3092\u7a7a\u3067\u306a\u3044\u5b8c\u5099\u8ddd\u96e2\u7a7a\u9593, $(F_n)_{n\\in \\mathbb{N}}$\u3092$X$ \u306e\u9589\u96c6\u5408\u304b\u3089\u306a\u308b\u5217\u3068\u3057, $X=\\bigcup_{n\\in \\mathbb{N}}F_n$\u3068\u3059\u308b.  \u3053\u306e\u3068\u304d\u3042\u308b$n\\in \\mathbb{N}$\u306b\u5bfe\u3057\u3066$F_n$\u306f\u5185\u70b9\u3092\u6301\u3064. \n\\end{cor}\n\\begin{proof}\n  \u4efb\u610f\u306e$n\\in \\mathbb{N}$\u306b\u5bfe\u3057\u3066$F_n$\u304c\u5185\u70b9\u3092\u6301\u305f\u306a\u3044\u3068\u4eee\u5b9a\u3059\u308b. \n  \u3053\u306e\u3068\u304d, $V_n:=X \\setminus F_n$ $(\\forall n\\in \\mathbb{N})$\u3068\u304a\u3051\u3070, $(V_n)_{n\\in \\mathbb{N}}$\u306f$X$\u306b\u304a\u3044\u3066\u7a20\u5bc6\u306a\u958b\u96c6\u5408\u304b\u3089\u306a\u308b\u5217\u3067\u3042\u308b. \n  \u5b9f\u969b, \u3042\u308b$x \\in X$\u3068$\\varepsilon > 0$\u306b\u5bfe\u3057\u3066$B(x,\\varepsilon) \\cap V_n = \\emptyset$\u3068\u3059\u308b\u3068$B(x,\\varepsilon) \\subset F_n$\u3068\u306a\u308a$F_n$\u304c\u5185\u70b9\u3092\u6301\u305f\u306a\u3044\u3053\u3068\u306b\u77db\u76fe\u3059\u308b. \n  \u3088\u3063\u3066, Baire\u306e\u30ab\u30c6\u30b4\u30ea\u5b9a\u7406\u3088\u308a$\\bigcap_{n\\in\\mathbb{N}}V_n$\u306f$X$\u306b\u304a\u3044\u3066\u7a20\u5bc6\u3067\u3042\u308b. \n  \u3068\u3053\u308d\u304c$\\bigcap_{n\\in \\mathbb{N}}V_n=X\\setminus \\bigcup_{n\\in \\mathbb{N}}F_n=\\emptyset$\u3067\u3042\u308b\u304b\u3089\u3053\u308c\u306f$X$\u304c\u7a7a\u3067\u306a\u3044\u3053\u3068\u306b\u77db\u76fe\u3059\u308b. \n\\end{proof}\n\n\\subsection{$C^{\\infty}$\u7d1a\u95a2\u6570\u304c\u591a\u9805\u5f0f\u3068\u4e00\u81f4\u3059\u308b\u305f\u3081\u306e\u5fc5\u8981\u5341\u5206\u6761\u4ef6}\n\u672c\u7bc0\u3067\u306f\u958b\u533a\u9593\u4e0a\u306e$C^{\\infty}$\u7d1a\u95a2\u6570\u304c$\\mathbb{R}$\u4fc2\u6570\u591a\u9805\u5f0f\u3068\u4e00\u81f4\u3059\u308b\u305f\u3081\u306e\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306b\u3064\u3044\u3066\u8ff0\u3079\u308b.  \u4ee5\u4e0b\u3067\u306f, \u3042\u308b\u958b\u533a\u9593\u4e0a\u3067\u5024\u304c\u4e00\u81f4\u3059\u308b\u4e8c\u3064\u306e$\\mathbb{R}$\u4fc2\u6570\u591a\u9805\u5f0f\u306f\u591a\u9805\u5f0f\u3068\u3057\u3066\u7b49\u3057\u3044\u3068\u3044\u3046\u4e8b\u5b9f\u3092\u65ad\u308a\u306a\u304f\u7528\u3044\u308b. \u3055\u3066, \u7c21\u5358\u306a\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u3068\u3057\u3066\u306f, \u3042\u308b\u6574\u6570$N \\geq 0$\u304c\u5b58\u5728\u3057\u3066$N$\u968e\u5fae\u5206\u304c\u6052\u7b49\u7684\u306b$0$\u3068\u306a\u308b\u3068\u3044\u3046\u3082\u306e\u304c\u3042\u308b\u304c, \u5b9f\u306f\u6b21\u306e\u5b9a\u7406\u304c\u6210\u7acb\u3059\u308b. \u306a\u304a, \u4ee5\u4e0b\u306e\u8a3c\u660e\u306f\\cite{Donoghue}\u3092\u53c2\u8003\u306b\u3057\u305f. \n\\begin{thm}\\label{ConditionOfSmoothFunctionIsPoly}\n  $f$\u3092\u958b\u533a\u9593$(c,d)$\u4e0a\u306e$C^{\\infty}$\u7d1a\u95a2\u6570\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u591a\u9805\u5f0f$p$\u304c\u5b58\u5728\u3057\u3066$(c,d)$\u4e0a$f=p$\u3068\u306a\u308b\u305f\u3081\u306e\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306f, \u4efb\u610f\u306e$x \\in (c,d)$\u306b\u5bfe\u3057\u3066\u3042\u308b\u6574\u6570$N_x \\geq 0$\u304c\u5b58\u5728\u3057\u3066$f^{(N_x)}(x)=0$\u3068\u306a\u308b\u3053\u3068\u3067\u3042\u308b. \n\\end{thm}\n\\begin{proof}\n  \u5fc5\u8981\u6027\u306f\u660e\u3089\u304b\u3067\u3042\u308b. \u5341\u5206\u6027\u3092\u793a\u3059. \n  \u5341\u5206\u5c0f\u3055\u3044\u3059\u3079\u3066\u306e$\\varepsilon > 0$\u306b\u3064\u3044\u3066\u3042\u308b\u591a\u9805\u5f0f$p$\u304c\u5b58\u5728\u3057\u3066$[c+\\varepsilon,d-\\varepsilon]$\u4e0a$f=p$\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044\u306e\u3067, \u306f\u3058\u3081\u304b\u3089$[c,d]$\u4e0a\u3067$C^{\\infty}$\u7d1a\u3068\u3057\u3066\u793a\u305b\u3070\u3088\u3044(\u7aef\u70b9\u3067\u306e\u5c0e\u5024\u306f\u7247\u5074\u5fae\u5206\u306e\u610f\u5473\u3067\u8003\u3048\u308b). \u958b\u96c6\u5408$G \\subset [c,d]$\u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057$F := [c,d] \\setminus G$\u3068\u304a\u304f. \n  \\[\n  G := \\{x \\in [c,d] \\mid x\\mbox{\u306e}[c,d]\\mbox{\u306b\u304a\u3051\u308b\u958b\u8fd1\u508d}U\\mbox{\u3068\u591a\u9805\u5f0f}p\\mbox{\u304c\u5b58\u5728\u3057\u3066}U\\mbox{\u4e0a}f=p \\}\n  \\]\n  $F = \\emptyset$\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044. \u5b9f\u969b, \u305d\u306e\u3068\u304d$G = [c,d]$\u3067\u3042\u308a, \u3042\u308b\u591a\u9805\u5f0f$p_1,p_2$\u3068$\\delta > 0$\u304c\u3042\u3063\u3066$[c,c+2\\delta)$\u4e0a$f=p_1$\u304b\u3064$(d-2\\delta,d]$\u4e0a$f=p_2$\u3068\u306a\u308b. \u305d\u3057\u3066, $a=c+\\delta, b= d - \\delta$\u3068\u304a\u304d, $[a,b] \\subset (c,d)$\u306e\u5404\u5143$x$\u306b\u5bfe\u3057\u3066$x$\u3092\u542b\u3080\u958b\u533a\u9593$I_x \\subset (c,d)$\u3068\u591a\u9805\u5f0f$p_x$\u3067$I_x$\u4e0a$f = p_x$\u306a\u308b\u3082\u306e\u3092\u53d6\u308b. \u3053\u306e\u3068\u304d$[a,b] \\subset \\bigcup_{x \\in [a,b]} I_x$\u3067\u3042\u308b\u306e\u3067$[a,b]$\u306e\u30b3\u30f3\u30d1\u30af\u30c8\u6027\u3088\u308a\u6709\u9650\u500b\u306e$x_1,\\ldots,x_n \\in [a,b]$\u304c\u5b58\u5728\u3057\u3066$[a,b] \\subset I_{x_1} \\cup \\cdots \\cup I_{x_n}$\u3068\u306a\u308b. \u305d\u3053\u3067$I_0 = [c,c+2\\delta),  I_1=I_{x_1}, \\ldots, I_n=I_{x_n}, I_{n+1} = (d-2\\delta,d]$\u3068\u304a\u304f\u3068$[c,d] = \\bigcup_{k=0}^{n+1} I_k$\u3068\u306a\u308b. \u3088\u3063\u3066$[c,d]$\u306e\u9023\u7d50\u6027\u3088\u308a\u591a\u9805\u5f0f\u3068\u3057\u3066\n  \\[\n  p_1 = p_{x_1}=\\cdots=p_{x_n}=p_2\n  \\]\n  \u304c\u308f\u304b\u308a$[c,d]$\u4e0a$f = p_1$\u3068\u306a\u308b. \u3088\u3063\u3066, $F = \\emptyset$\u3092\u793a\u305b\u3070\u3088\u3044. \u4ee5\u4e0b\u3067\u306f$F \\neq \\emptyset$\u3068\u4eee\u5b9a\u3057\u3066\u77db\u76fe\u3092\u5c0e\u304f. \\\\\n  {\\bf(Step1)}:$F$\u304c\u5b64\u7acb\u70b9\u3092\u6301\u305f\u306a\u3044\u3053\u3068\u3092\u793a\u3059. $x_0 \\in F$\u304c\u5b64\u7acb\u70b9\u3067\u3042\u308b\u3068\u3057\u3066\u77db\u76fe\u3092\u5c0e\u304f. \\\\\n  $(1)$:$x_0 = d$\u306e\u5834\u5408. \n  \u3042\u308b$b \\in (c,d)$\u304c\u5b58\u5728\u3057\u3066$(a,x_0) \\subset G$\u3068\u306a\u308b. \n  \u3053\u306e\u3068\u304d, \u3042\u308b$\\delta > 0$\u3068\u591a\u9805\u5f0f$P$\u304c\u5b58\u5728\u3057\u3066$(a,a+\\delta) \\subset (a,x_0)$\u4e0a$f=P$\u3068\u306a\u308b. \u305d\u3053\u3067\n  \\[\n  y_0 := \\sup\\{y \\in (a,x_0] \\mid \\forall x \\in (a,y),~ f=P \\}\n  \\]\n  \u3068\u304a\u304f\u3068$y_0 = x_0$\u3068\u306a\u308b. \n  \u5b9f\u969b, $y_0 < x_0$\u3068\u3059\u308b\u3068$y_0 \\in (a,x_0) \\subset G$\u3067\u3042\u308b\u306e\u3067, \u3042\u308b$\\varepsilon>0$\u3068\u591a\u9805\u5f0f$Q$\u304c\u5b58\u5728\u3057\u3066$(y_0-\\varepsilon,y_0+\\varepsilon) \\subset (a,x_0]$\u4e0a\u3067$f=Q$\u3068\u306a\u308b.\n  \u4e00\u65b9, $y_0$\u306e\u5b9a\u7fa9\u304b\u3089$y-\\varepsilon<z\\leq y_0$\u3067$(a,z)$\u4e0a\u3067$f=P$\u3068\u306a\u308b\u3082\u306e\u304c\u53d6\u308c\u308b. \n  \u5f93\u3063\u3066, $(y-\\varepsilon,z)$\u4e0a\u3067$P=f=Q$\u3068\u306a\u308b\u306e\u3067\u591a\u9805\u5f0f\u3068\u3057\u3066$P=Q$\u3067\u3042\u308b. \u3088\u3063\u3066, $(a,y_0+\\varepsilon)$\u4e0a\u3067$f=Q$\u3068\u306a\u308b\u304c\u3053\u308c\u306f$y_0$\u306e\u5b9a\u7fa9\u306b\u53cd\u3059\u308b. \u4ee5\u4e0a\u304b\u3089$(a,x_0]$\u4e0a\u3067$f=P$\u3068\u306a\u308b. \u3088\u3063\u3066, $(a.x_0]$\u306f$x_0$\u306e$[c,d]$\u306b\u304a\u3051\u308b\u958b\u8fd1\u508d\u3067\u3042\u308b\u306e\u3067$x_0 \\in G$\u3068\u306a\u308b. \u3053\u308c\u306f\u4e0d\u5408\u7406. \\\\\n  $(2)$:$x_0=c$\u306e\u5834\u5408. $(1)$\u3068\u540c\u69d8\u306b\u77db\u76fe\u304c\u5c0e\u304b\u308c\u308b. \\\\\n  $(3)$:$x_0 \\in (c,d)$\u306e\u5834\u5408. \n  \u3042\u308b$a,b \\in(c,d)$\u304c\u5b58\u5728\u3057\u3066$[a,x_0), (x_0,b] \\subset G$\u3068\u306a\u308b. \n  \u3053\u306e\u3068\u304d, $(1)$\u306e\u8a3c\u660e\u3068\u540c\u69d8\u306b\u3042\u308b\u591a\u9805\u5f0f$P$\u304c\u5b58\u5728\u3057\u3066$(a,x_0]$\u4e0a\u3067$f=P$\u3068\u306a\u308b. \u307e\u305f, \u3042\u308b\u591a\u9805\u5f0f$P'$\u304c\u5b58\u5728\u3057\u3066$[x_0,b)$\u4e0a\u3067$f=P'$\u3068\u306a\u308b. \u3053\u306e\u3068\u304d$P=P'$\u3067\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044. \n  \u3053\u306e\u3053\u3068\u3092\u793a\u3059\u305f\u3081\u306b\u307e\u305a, $n:=\\mathrm{deg}(P) = \\mathrm{deg}(P') =: m$\u3092\u793a\u3059. \n  \u3044\u307e$(a,x_0]$\u4e0a\u3067$f$\u306f$n$\u6b21\u591a\u9805\u5f0f\u306a\u306e\u3067$f^{(k)}(x_0)=0 ~(k \\geq n+1)$\u3067\u3042\u308b. \u4e00\u65b9, $[x_0,b)$\u4e0a\u3067$f$\u306f$m$\u6b21\u591a\u9805\u5f0f\u306a\u306e\u3067$f^{(m)}(x_0) \\neq 0$\u3067\u3042\u308b. \u305d\u3053\u3067\u3082\u3057$n < m$\u306a\u3089\u3070$0 = f^{(m)}(x_0) \\neq 0$\u3068\u306a\u308b. \u3053\u308c\u306f\u4e0d\u5408\u7406\u3067\u3042\u308b\u306e\u3067$n \\geq m$\u3067\u3042\u308b. \u540c\u69d8\u306b$n \\leq m$\u306a\u306e\u3067$n=m$\u3067\u3042\u308b. \u6b21\u306b$x_0$\u4e2d\u5fc3\u306e$f$\u306eTaylor\u5c55\u958b\u3092\u8003\u3048\u308b\u3068\n  \\[\n  \\begin{aligned}\n  &P(x) = f(x) = \\sum_{k=0}^n \\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k ~~~(x \\in (a,x_0]) \\\\\n  &P'(x) = f(x) = \\sum_{k=0}^n \\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k ~~~(x \\in [x_0,b)) \n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b. \u3053\u308c\u3088\u308a\n  \\[\n  \\begin{aligned}\n  &P(x) = \\sum_{k=0}^n \\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k ~~~(x \\in \\mathbb{R}) \\\\\n  &P'(x) = \\sum_{k=0}^n \\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k ~~~(x \\in \\mathbb{R})\n  \\end{aligned}\n  \\]\n  \u3068\u306a\u308b\u304b\u3089$P=P'$\u3067\u3042\u308b. \u4ee5\u4e0a\u304b\u3089$(a,b)$\u4e0a\u3067$f=P$\u3067\u3042\u308b. \u3053\u308c\u306f$x_0 \\in G$\u3092\u610f\u5473\u3059\u308b. \u3068\u3053\u308d\u304c$x_0 \\in F$\u306a\u306e\u3067\u3053\u308c\u306f\u4e0d\u5408\u7406. \u3088\u3063\u3066$F$\u306f\u5b64\u7acb\u70b9\u3092\u6301\u305f\u306a\u3044. \\\\\n  {\\bf(Step2)}:\u6574\u6570$N \\geq 0$\u3068\u9577\u3055\u6b63\u306e\u9589\u533a\u9593$I \\subset (c,d)$\u304c\u5b58\u5728\u3057\u3066$F \\cap I \\neq \\emptyset$\u304b\u3064$F \\cap I$\u4e0a$f^{(k)}=0 ~(\\forall k \\geq N)$\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3059. \n  \u6574\u6570$n \\geq 0$\u306b\u5bfe\u3057\u3066$E_n := \\{x \\in F \\mid f^{(n)}(x) = 0\\}$\u3068\u304a\u304f. $E_n$\u306f$F$\u306e\u9589\u96c6\u5408\u3067\u3042\u308a, \u5b9a\u7406\u306e\u4eee\u5b9a\u3088\u308a$F = \\bigcup_{n=0}^{\\infty} E_n$\u3067\u3042\u308b. \u3044\u307e$F \\neq \\emptyset$\u306f\u9589\u533a\u9593$[c,d]$\u306e\u9589\u96c6\u5408\u306a\u306e\u3067\u5b8c\u5099\u8ddd\u96e2\u7a7a\u9593\u3067\u3042\u308b. \u3088\u3063\u3066, Baire\u306e\u30ab\u30c6\u30b4\u30ea\u30fc\u5b9a\u7406(\u7cfb\\ref{BaireCategoryTheorem})\u3088\u308a\u3042\u308b\u6574\u6570$N \\geq 0$\u304c\u5b58\u5728\u3057\u3066$E_N$\u306f\u5185\u70b9\u3092\u6301\u3064. \u3086\u3048\u306b\u9577\u3055\u6b63\u306e\u9589\u533a\u9593$I \\subset (c,d)$\u304c\u5b58\u5728\u3057\u3066$\\emptyset \\neq F \\cap I \\subset E_N$\u3068\u306a\u308b. Step$1$\u3088\u308a$F$\u306f\u5b64\u7acb\u70b9\u3092\u6301\u305f\u306a\u3044\u306e\u3067$F \\cap I$\u3082\u5b64\u7acb\u70b9\u3092\u6301\u305f\u306a\u3044. \u3086\u3048\u306b$x \\in F \\cap I$\u306b\u5bfe\u3057\u3066$x_n \\rightarrow x$\u306a\u308b$(F \\cap I) \\setminus \\{x\\}$\u306e\u5143\u304b\u3089\u306a\u308b\u5217$(x_n)_{n \\in \\mathbb{N}}$\u304c\u53d6\u308c\u308b\u306e\u3067, \n  \\[\n  f^{(N+1)}(x) = \\lim_{n \\rightarrow \\infty} \\frac{f^{(N)}(x_n) - f^{(N)}(x)}{x_n - x} = \\lim_{n \\rightarrow \\infty} \\frac{0 - 0}{x_n - x} = 0\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066\u5e30\u7d0d\u7684\u306b$F \\cap I$\u4e0a$f^{(k)}=0 ~(\\forall k \\geq N)$\u3068\u306a\u308b. \\\\\n  {\\bf (Step3)}:Step2\u306e$N$\u3068$I$\u306b\u3064\u3044\u3066$I$\u4e0a\u3067$f^{(N)}=0$\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3059. \u3053\u308c\u304c\u793a\u3055\u308c\u308c\u3070$I \\subset G$\u3067\u3042\u308b\u306e\u3067$\\emptyset \\neq F \\cap I = \\emptyset$\u306a\u308b\u77db\u76fe\u306b\u5230\u9054\u3059\u308b. \n  \u3055\u3066, Step2\u3088\u308a$F \\cap I$\u4e0a\u3067$f^{(N)}=0$\u3067\u3042\u308b\u306e\u3067$G \\cap I \\neq \\emptyset$\u3068\u3057\u3066$G \\cap I$\u4e0a\u3067$f^{(N)}=0$\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044. \n  \u4efb\u610f\u306b$x_0 \\in G \\cap I$\u3092\u3068\u308b. $x_0$\u3092\u542b\u3080\u958b\u533a\u9593$(a,b) \\subset G \\cap I$\u3068\u591a\u9805\u5f0f$p$\u304c\u5b58\u5728\u3057\u3066$(a,b)$\u4e0a$f=p$\u3068\u306a\u308b. \u3053\u3053\u3067\n  \\[\n  a_0 = \\inf \\{ y \\in I \\mid \\forall x \\in (y,b),~ f(x)=p(x) \\}\n  \\]\n  \u3068\u304a\u304f. \u3059\u308b\u3068, $a_0 \\in F \\cap I$\u3067\u3042\u308b. \n  \u5b9f\u969b, $a_0 \\in G \\cap I$\u3067\u3042\u308b\u306a\u3089\u3070, $I \\subset (c,d)$\u3088\u308a$a_0$\u306f\u7aef\u70b9\u3067\u306f\u306a\u3044\u306e\u3067, \u591a\u9805\u5f0f$p'$\u3068$\\delta>0$\u304c\u5b58\u5728\u3057\u3066$(a_0-\\delta,a_0+\\delta) \\subset G \\cap I$\u4e0a\u3067$f=p'$\u3068\u306a\u308b. \u3059\u3050\u308f\u304b\u308b\u3088\u3046\u306b$(a_0,a_0+\\delta)$\u4e0a\u3067$p'=f=p$\u3067\u3042\u308b\u306e\u3067\u591a\u9805\u5f0f\u3068\u3057\u3066$p=p'$\u3068\u306a\u308a, \u5f93\u3063\u3066, $(a_0-\\delta,b)$\u4e0a\u3067$f=p$\u3068\u306a\u308b. \u3053\u308c\u306f$a_0$\u306e\u5b9a\u7fa9\u306b\u53cd\u3059\u308b. \u3088\u3063\u3066$a_0 \\in F \\cap I$\u3067\u3042\u308b. \n  \u3086\u3048\u306b, Step2\u3088\u308a$f^{(k)}(a_0)=0 ~(\\forall k \\geq N)$\u3068\u306a\u308b. \n  \u4e00\u65b9, $(a_0,b)$\u4e0a\u3067$f=p$\u3067\u3042\u308b\u306e\u3067$\\mathrm{deg}(p) = m$\u3068\u304a\u304f\u3068, \n  \\[\n  f(z) = \\sum_{k=0}^m \\frac{f^{(k)}(a_0)}{k!}(z-a_0)^k ~~~(z \\in (a_0,b))\n  \\]\n  \u3068\u306a\u308b. \u3053\u308c\u3068$f^{(k)}(a_0)=0 ~(\\forall k \\geq N)$\u3092\u5408\u308f\u305b\u308b\u3068\n  \\[\n  f(z) = \\sum_{k=0}^{N-1} \\frac{f^{(k)}(a_0)}{k!}(z-a_0)^k ~~~(z \\in (a_0,b))\n  \\]\n  \u3068\u306a\u308b. \u3088\u3063\u3066, $f^{(N)}(x_0) = 0$\u3067\u3042\u308b. \n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDefect and texture engineering of soft matter~\\cite{jangizehi2020defects,nielsen2020substrate} are promising design approaches for tuning the properties of materials by controlling the presence and spatial distribution of defects. Modulated soft phases, such as block copolymers~\\cite{re:ruzette05}, smectic liquid crystals~\\cite{degennes1995physics}, active and living matter~\\cite{re:mitov17}, are of particular interest in this context since their broken translational symmetry is associated with lamellar patterns of uniaxial symmetry, which allows for a variety of topological defects. We focus here on smectic thin films which are known, under appropriate boundary conditions, to form focal conic domains: topological defects that arrange themselves in periodic arrays throughout the film, and form due to the hybrid molecular alignment between a substrate and the film's free surface.\n\nThe smectic phase of a liquid crystal is comprised of anisometric molecules that present collective orientational order along with a director axis $\\mathbf{n}$, and are organized in periodically spaced molecular layers. When deposited on a substrate that induces tangential anchoring, smectic films in contact with air (with homeotropic molecular alignment at the smectic-air interface) may have their layers bend and form focal conic domains (FCDs)~\\cite{shojaei2006role,guo2008controlling,kleman2009liquids,kim2018curvatures}, depending on the balance between elastic energy and surface anchoring~\\cite{lavrentovich1994nucleation,kim2009confined}. These conical defects have been long known to be the equilibrium structure of the film, since the works of Friedel and Bragg in the  1920s and 1930s~\\cite{friedel1922etats,bragg1934liquid}. The external layer forms a cusp-like macroscopic singularity at its center, whose neighborhood is a region of high mean and Gaussian curvatures. The mean curvature $H = (c_1+c_2)\/2$ is the average of the principal curvatures $c_1 = 1\/R_1$ and $c_2 = 1\/R_2$ at a surface point (inverse of the principal radii), while the Gaussian curvature $G = c_1 c_2$ is the product of these curvatures. Figure~\\ref{fig:fc-radius} presents a schematic for a toroidal FCD (axially symmetric) with multiple layers, defined within a cylindrical region of radius \\textsl{a} embedded into a smectic matrix of flat layers. The principal radius $R_1$ of a point on a layer ($c_1 > 0$), associated with the hybrid alignment, has its origin at the bottom of the cylinder defined by \\textsl{a}, whereas $R_2$ is antiparallel to $R_1$ ($c_2< 0$), and connects this point to the axis of symmetry of the FCD. One can observe that $R_2$ of the external molecular layer becomes very small as the core is approached.\n\n\\begin{figure}[ht]\n\t\\centering\n    \\includegraphics[width=0.42\\textwidth]{figures\/fc-radius.pdf}\n    \\caption{Structure of a toroidal FCD within a cylindrical region of radius \\textsl{a}, presenting multiple molecular layers embedded in a smectic matrix of flat layers~\\cite{kim2016controlling}. $R_1$ (with respective $c_1 > 0$) and $R_2$ ($c_2<0$) are the principal radii of a point on the external layer, so that the mean curvature $H$ changes from positive to negative as the core is approached, while the Gaussian curvature $G$ is negative everywhere.}\n\t\\label{fig:fc-radius}\n\\end{figure}\n\nSmectic films micropatterned with arrays of FCDs have been used as guides for colloidal dispersion~\\cite{milette2012reversible,pratibha2010colloidal}, soft lithography~\\cite{yoon2007internal,kim2010self}, and as templates for superhydrophobic surfaces~\\cite{kim2014creation}. Nevertheless, there are still open problems associated with the formation and engineering uses of these defects. Recent findings have shown, for example, the existence of geometric memory in the nematic-smectic transition~\\cite{gim2017morphogenesis,suh2019topological}, connecting the presence of boojum defects in the nematic to the nucleation of focal conics in the smectic, and have also shown that focal conics may even arise in the absence of hybrid alignment~\\cite{selmi2017structures}. While the nucleation of defects in smectics has appealing functional applications, focal conics and dislocations can strongly dictate the structure of easily deformable thin films~\\cite{coursault2016self}, and interfere in optical and conductivity properties~\\cite{jangizehi2020defects}. Therefore, fine microstructural control is a key concern for soft material design.\n\nA remarkable example is found in the experiments of Kim et al.~\\cite{kim2016controlling,kim2018curvatures}, who have shown that smectic thin films presenting arrays of FCDs can undergo complex morphological transitions through sintering (i.e. reshaping of a smectic film at elevated temperatures for a certain amount of amount, with subsequent cooling). For particular sintering protocols, FCDs are sculpted through evaporation into unexpected patterns, which include conical pyramids and concentric rings. The observed configurations are controlled not only by the local mean curvature of the film surface (as classical thermodynamics would imply), but also by its Gaussian curvature~\\cite{kim2016controlling}. This feature, together with the fact that the film is constantly evaporating and reshaping at elevated temperatures, are the main motivations for our study. We develop a three-dimensional phase-field framework that includes smectic elasticity in the film, interfacial thermodynamics and kinetics accounting for the effects of Gaussian curvature, and material compressibility at the smectic-air surface. \n\nThe Oseen-Frank theory~\\cite{oseen1933theory,frank1958liquid} of a bulk nematic phase naturally leads to incorporating the energy associated with small layer distortions as a function of a layer displacement variable in a bulk smectic~\\cite{degennes1995physics,santangelo2005curvature}. Layer displacements along their normal direction are assumed parallel to the nematic director since their difference is a higher energy mode that relaxes quickly~\\cite{santangelo2005curvature} (a constraint that may be abandoned with additional balance equations~\\cite{capriz2001swelling}). This approach, alongside an explicit curvature elastic energy description~\\cite{kleman2000curvature} adequate for surfaces with large mean and Gaussian curvatures, such as FCDs, have been successful in explaining why these defects are equilibrium configurations, in which layers are geometrically described as Dupin cyclides~\\cite{friedel1922etats,schief2005nonlinear}. A nonlinear theory coupling the director to hydrodynamic flow was formulated by Brand and Pleiner~\\cite{brand1980nonlinear}, which expanded on previous studies on flows and viscosities in mesophases~\\cite{leslie1966some,martin1972unified,moritz1976nonlinearities}. A complete hydrodynamic theory for studying the smectic-isotropic transition was later proposed based on the Landau-de Gennes framework~\\cite{brand2001macroscopic,mukherjee2001simple,abukhdeir2010edge}, written in terms of either the smectic layer displacement, the smectic complex order parameter, or the nematic $Q-$tensor. Small perturbations in a smectic are generally described by a scalar displacement normal to the layers. However, the layering can be also described by a complex amplitude, the leading order term in a Fourier series of the order parameter, with a phase that is a function of layer displacement, and a real amplitude that represents the strength of the local smectic order. Other coarse-grained models along these lines have also been proposed, based either on density deviations or local composition~\\cite{poniewierski1991phase,linhananta1991phenomenological,pevnyi2014modeling,pezzutti2015smectic,xia2021structural}.\n\nWe have introduced a phase-field model for a smectic-isotropic system~\\cite{vitral2019role,vitral2020model}, with a real order parameter $\\psi$ that describes a modulated smectic phase in contact with an isotropic phase, and the diffuse interface between the two. By replacing discrete layers and describing two-phase interfaces by a smooth function, we allow tracking of arbitrarily distorted smectic planes, even while layers are breaking up or forming, as well as allowing two phase interfaces with cusps (such as in FCDs). We have chosen the energy of our phase-field model so that the Oseen-Frank energy for a smectic is recovered for small distortions; that is, the energy penalizes both deviations from the equilibrium layering spacing and bending of layers (splay of molecules). An asymptotic analysis of the model for a thin two phase interface reproduces the classical Gibbs-Thomson relation of interfacial thermodynamics, as well as corrections introduced by higher order curvatures and torsion. This extended thermodynamic relation at a distorted smectic-isotropic interface yields modified kinetic equations that account for the role of the Gaussian curvature and layering orientation on pattern formation, providing an explanation for the obervations by Kim et al.~\\cite{kim2016controlling}. Numerical solutions of the governing equations reveal transitions from a FCD towards conical pyramids slightly away from coexistence (simulating a temperature increase in the experiments). \n\nThe model of~\\cite{vitral2019role} is limited by the assumption of incompressibility, and neglects advection of the modulated order parameter. That is, it strictly describes the diffusive evolution of an interface between a smectic and an isotropic phase of the same, constant, density. As our goal is to model an isotropic phase of lower density than the smectic, that is, an isotropic fluid (such as water or air) presenting small density of liquid crystal molecules, in~\\cite{vitral2020model} we presented a way to incorporate a varying density field and hydrodynamics into the model. Smectic and isotropic phases of different density were considered, and fluid flows modeled along the lines of Cahn-Hilliard fluids~\\cite{re:jasnow96,gurtin1996two} as extended by Lowengrub and Truskinovsky's to quasi-incompressible fluids~\\cite{lowengrub1998quasi} (see also \\cite{lee2002modeling,abels2012thermodynamically,guo2017mass,gong2018fully,ding2007diffuse,shokrpour2018diffuse}). This assumption led to a density which smoothly varied from a uniform value inside the bulk of the smectic ($\\rho_s$) to a uniform value in the bulk of the isotropic phase ($\\rho_0$). However, the density and the smectic order parameter were not considered to be independent variables, but rather the local density was assumed to be a constitutive function of the local amplitude of the order parameter (quasi-incompressibility).  In this scenario, the balance of mass presents a strong constraint on how much the non-conserved order parameter can evolve. Therefore, the morphological transitions from FCDs we previously discussed~\\cite{vitral2019role,kim2016controlling} are unable to take place in a quasi-incompressible smectic-isotropic system, so that the range of applicability of the model is limited.\n\nIn this work, we overcome these limitations by introducing a general model in which the smectic order parameter and the density are treated as independent variables, with model parameters that control how constrained is the smectic evolution, allowing for morphological transitions. We call it the \\textit{weakly compressible smectic-isotropic model}.  We add a coupling (or penalty) term to the energy that limits deviations of the local density from the expected smectic and isotropic equilibrium densities in the bulk phases. The model enforces mass conservation, allows control on how strongly the conserved density affects the motion of the non-conserved smectic order parameter, and is shown to be numerically stable even in presence of a large density ratio between phases. This in sharp contrast with previously studied phase-field models for a modulated phase in contact with a $\\psi = 0$ phase~\\cite{thiele2013localized,knobloch2015spatial,espath2020generalized}, in which the order parameter $\\psi$ was (i) non-conserved and free to evolve (e.g. Swift-Hohenberg dynamics) or (ii) conserved and restricted in its evolution (e.g. phase-field-crystal models). The weakly compressible smectic-isotropic model allows for non-conserved order parameter dynamics with a conserved density, as is physically the case. \n\nThe paper is organized as follows. Section~\\ref{sec:wcm} contains the model derivation, including the equations for the order parameter and balances of mass and momentum. We introduce a rotationally invariant energy which depends on a real order parameter $\\psi$ and its gradients, with an additional penalty term for deviations from equilibrium density values. In Sec.~\\ref{sec:ns} we discuss our numerical implementation based on a pseudo-spectral method. Section~\\ref{sec:def} presents numerical results of growth and decay of planar smectic layers in order to show how the conserved density interacts with the non-conserved order parameter. In Sec.~\\ref{sec:fcflow} we reconsider the evolution of FCDs, and show that morphological transitions from focal conics to conical pyramids or concentric rings can be obtained numerically from the weakly compressible model as long as the density coupling coefficient is not large. We also examine the flow at the surface of a FCD and a conical pyramid, and, based on the extended Gibbs-Thomson relation, we discuss how interfacial flows and stresses depend not only on the mean curvature but on the Gaussian curvature and layering orientation as well. Finally, in Sec.~\\ref{sec:domain} we consider additional applications of the weakly compressible model, namely the coalescence of cylindrical stacks of smectic layers, formation of droplets from an initial smectic, and the interactions among neighboring FCDs in a smectic film.\n\n\\section{Weakly Compressible model}\n\\label{sec:wcm}\n\nIn this section we derive a diffuse interface model for a weakly compressible smectic phase in contact with an isotropic fluid when they have different equilibrium densities. A real order parameter representing the smectic layering is introduced~\\cite{chaikin2000principles},\n\\begin{equation}\n    \\psi = \\sum_n \\frac{1}{2}[\\bar{A}_n \\,e^{in \\mathbf{q}_0\\cdot\\mathbf{x}}+c.c.] \\;,\n\\end{equation}\nwhere $\\mathbf{q}_0$ is a wave vector in the direction normal to the smectic layers, $q_0 = |\\mathbf{q}_0|$ is its magnitude, $\\lambda_0 = 2\\pi\/q_0$ is the characteristic period, and $\\bar{A}_n$ is a complex amplitude. The order parameter $\\psi$ is function of time $t$ and space $\\mathbf{x} \\in \\mathbb{R}^3$. Near the transition, high harmonics are generally negligible, and smectic layering is well described by the approximate representation $\\psi \\approx \\frac{1}{2}[\\bar{A}\\,e^{i\\mathbf{q}_0\\cdot\\mathbf{x}}+c.c.]$. The complex amplitude has the form $\\bar{A} = A\\,e^{-iq_0 u}$, where $u(\\mathbf{x},t)$ represents the displacement away from planar smectic layers, and $A$ is the real magnitude of the complex amplitude (the order parameter \\textit{strength}).\n\nWe write the internal energy $\\mathfrak{U}$ of the system in terms of the energy per unit mass $\\mathfrak{u}$ and the mass density $\\rho$, where $\\mathfrak{u} = \\mathfrak{u}(\\mathfrak{s},\\psi,\\nabla \\psi,\\nabla^2\\psi)$ and $\\mathfrak{s}$ is the specific internal entropy. The functional dependence on $\\nabla^2\\psi$ does not appear for binary systems~\\cite{gurtin1996two,lowengrub1998quasi}, but it is fundamental to model the smectic phase, as it makes the energy sensitive to layer distortions and curvatures. We also introduce in the energy an explicit coupling between the real amplitude $A$ of the modulated order parameter $\\psi$, which is approximately constant in the bulk smectic and isotropic phases, and the density of the corresponding phase. This internal energy is written as\n\\begin{eqnarray}\n    \\mathfrak{U} &=& \\int_\\Omega \\bigg\\{\\rho \\mathfrak{u}\\, + \\frac{\\zeta}{2}\\Big[\\rho-\\rho_0-\\kappa A\\Big]^2\\bigg\\} d\\mathbf{x} \\; .\n    \\label{eq:energy}\n\\end{eqnarray}\nThe second term inside the integral penalizes density values that deviate from equilibrium values $\\rho_{eq} = \\rho_0+\\kappa A$ in the smectic and isotropic phases, where $\\zeta$, $\\rho_0$ (the equilibrium density for the isotropic phase) and $\\kappa$ are constants. In the limit of $\\zeta \\rightarrow \\infty$, the density becomes constitutively governed by $A$, as $\\rho = \\rho_0 + \\kappa A$, and the energy reduces to $\\mathfrak{U} = \\int_\\Omega \\rho\\,\\mathfrak{u}\\,dx$. This is the quasi-incompressible limit that we previously studied in~\\cite{vitral2020model}, and it imposes a strict constraint on the evolution of the non-conserved order parameter $\\psi$. However, when $\\zeta$ is finite, the density is an independent variable.\n\n\\subsection{Energy and entropy balances}\n\nWe first obtain the local form of the internal energy and entropy balances from the fist law of Thermodynamics, as detailed in~\\cite{lowengrub1998quasi,vitral2020model}. The relations derived in this section are obtained in the absence of thermal radiation and heat flux though the boundary. When deriving the governing equations, we set no-flux boundary conditions: Neumann condition for the order parameter $\\psi$ and zero normal fluid velocity $\\mathbf{v}$ on the boundary, such that\n\\begin{eqnarray}\n    \\nabla \\psi (\\mathbf{x})  \\cdot \\mathbf{n} = \\nabla \\nabla^2 \\psi (\\mathbf{x})  \\cdot \\mathbf{n} =  0, \\quad \\mathbf{v}(\\mathbf{x})\\cdot \\mathbf{n} = 0, \\quad \\mathbf{x} \\in \\partial \\Omega.\n\\label{eq:bc-w}\n\\end{eqnarray}\nGiven the Neumann condition for $\\psi$, the wave vector $\\mathbf{q}_0$ is parallel to the boundaries, so that layers become perpendicularly anchored in these regions, allowing for focal conics to be created. \n\nWe introduce $\\mathbf{T}$ as the Cauchy stress tensor, and define the material time derivative of a vector $\\mathbf{g}$ as $\\dot{\\mathbf{g}} = \\partial_t\\mathbf{g}+\\mathbf{v}\\cdot\\nabla\\mathbf{g}$. The first law is then stated as\n\\begin{eqnarray}\n    \\frac{d}{dt}\\int_\\Omega\\bigg\\{\\rho \\mathfrak{u}\\, + \\frac{\\zeta}{2}\\Big[\\rho-\\rho_0-\\kappa A\\Big]^2 + \\frac{\\rho|\\mathbf{v}|^2}{2}\\bigg\\}d\\mathbf{x}\n    = \\int_{\\partial\\Omega}\\bigg[\\mathbf{T}\\mathbf{n}\\cdot\\mathbf{v}+(\\boldsymbol\\xi\\cdot\\mathbf{n})\\dot{\\psi}\\bigg]dS \\;,\n\\end{eqnarray}\nwhere the surface integral is the rate of work done on the surface of the system, and the volume integral includes both internal energy and kinetic energy.\n\nGiven the balances of linear momentum and mass,\n\\begin{equation}\n    \\rho\\dot{\\mathbf{v}}=\\nabla\\cdot\\mathbf{T} \\quad,\\quad \n    \\dot{\\rho}+\\rho\\nabla\\cdot\\mathbf{v}=0 \\;,\n\\end{equation}\nthe local form of the balance of internal energy~\\cite{gurtin2010mechanics} becomes\n\\begin{eqnarray}\n    \\rho\\dot{\\mathfrak{u}}-\\kappa\\zeta\\Big(\\rho-\\rho_0-\\kappa A\\Big)\\dot{A} &=& \\bigg\\{\\mathbf{T}+\\frac{\\zeta}{2}\\bigg[\\rho^2-(\\rho_0+\\kappa A)^2\\bigg]\\bigg\\}:\\nabla \\mathbf{v} + \\nabla \\cdot (\\boldsymbol\\xi\\,\\dot{\\psi}) \\;.\n    \\label{eq:ebal-local-w}\n\\end{eqnarray}\nHere $\\boldsymbol\\xi$ is a microstress (also known as a generalized force), whose origin lies in the theory of microforces, a generalization of order parameter theories such as Ginzburg-Landau and Cahn-Hilliard~\\cite{hutter2020coleman,espath2020generalized,vitral2020mesoscale,duda2021coupled}. Similarly to the balance of linear momentum, we can write the balance of microforces in terms of the microforce $\\pi$ as\n\\begin{equation}\n    \\nabla\\cdot\\boldsymbol\\xi + \\pi = 0 \\;.\n\\end{equation}\n\nSince the energy density is of the form $\\mathfrak{u} = \\mathfrak{u}(\\mathfrak{s},\\psi,\\nabla \\psi,\\nabla^2\\psi)$, from partial differentiation the local balance of entropy can be derived from Eq.~(\\ref{eq:ebal-local-w}). Note that $\\mathfrak{u}$ is an energy per unit mass, independent of the density $\\rho$. Therefore, by the chain rule\n\\begin{equation}\n    \\dot{\\mathfrak{u}} =\n        \\frac{\\partial \\mathfrak{u}}{\\partial \\mathfrak{s}}\\dot{\\mathfrak{s}} \n        + \\frac{\\partial \\mathfrak{u}}{\\partial \\psi}\\dot{\\psi}\n        + \\frac{\\partial \\mathfrak{u}}{\\partial \\nabla\\psi}\\cdot\\dot{\\overline{\\nabla\\psi}}\n        + \\frac{\\partial \\mathfrak{u}}{\\partial \\nabla^2\\psi}\\dot{\\overline{\\nabla^2\\psi}} \\; .\n        \\label{eq:chain}\n\\end{equation}\nBy accounting for the boundary conditions from Eq.~(\\ref{eq:bc-w}), the gradient terms appearing in $\\rho\\dot{\\mathfrak{u}}$ can be rewritten as\n\\begin{eqnarray}\n \\rho \\frac{\\partial \\mathfrak{u}}{\\partial \\nabla\\psi}\\cdot\\dot{\\overline{\\nabla\\psi}} &=& \n \\rho\\frac{\\partial \\mathfrak{u}}{\\partial \\nabla\\psi}\\cdot \\nabla\\dot{\\psi} \n - \\rho \\nabla\\psi \\otimes \\frac{\\partial \\mathfrak{u}}{\\partial\\nabla\\psi} : \\nabla\\mathbf{v} \\; ,\n\\end{eqnarray}\nand also\n\\begin{eqnarray}\n    \\nonumber\n    \\rho \\frac{\\partial \\mathfrak{u}}{\\partial \\nabla^2\\psi}\\dot{\\overline{\\nabla^2\\psi}} &=&\n    \\rho \\frac{\\partial \\mathfrak{u}}{\\partial \\nabla^2\\psi}\\nabla^2\\dot{\\psi} \n    - \\rho \\frac{\\partial \\mathfrak{u}}{\\partial \\nabla^2\\psi}\\nabla^2\\mathbf{v}\\cdot\\nabla\\psi\n    -2\\rho \\frac{\\partial \\mathfrak{u}}{\\partial \\nabla^2\\psi}\\nabla\\mathbf{v} : \\mathbf{D}\\psi \n   \\\\[2mm] &=&\n    -\\nabla\\bigg(\\rho\\frac{\\partial \\mathfrak{u}}{\\partial\\nabla^2\\psi}\\bigg)\\cdot\\nabla\\dot{\\psi}\n    + \\bigg[ \\nabla\\psi\\otimes\\nabla\\bigg(\\rho \\frac{\\partial \\mathfrak{u}}{\\partial\\nabla^2\\psi}\\bigg)\n    - \\rho \\frac{\\partial \\mathfrak{u}}{\\partial\\nabla^2\\psi} \\mathbf{D}\\psi \\bigg] : \\nabla\\mathbf{v} \\; ,\n\\end{eqnarray}\nwhere $\\mathbf{D}$ stands for $\\nabla\\nabla$ (i.e. $``\\partial_i\\partial_j \"$), so that $\\mathbf{D}\\psi$ is a second order tensor. \n\nGiven that the temperature $\\theta = \\partial \\mathfrak{u}\/ \\partial \\mathfrak{s}$ and that the real amplitude of the order parameter has a simple dependency $A = A(\\psi)$, by substituting Eq.~(\\ref{eq:chain}) into Eq.~(\\ref{eq:ebal-local-w}) we obtain the following local balance of entropy:\n\\begin{eqnarray}\n    \\nonumber\n    \\rho \\theta \\dot{\\mathfrak{s}} &=&\n   \n        \\bigg\\{ \\mathbf{T} + \\frac{\\zeta}{2}\\Big[\\rho^2-(\\rho_0+\\kappa A)^2 \\Big]\\mathbf{I}\n        + \\rho \\nabla\\psi \\otimes \\frac{\\partial \\mathfrak{u}}{\\partial\\nabla\\psi}\n        -\\nabla\\psi\\otimes\\nabla\n        \\bigg(\\rho\\frac{\\partial \\mathfrak{u}}{\\partial \\nabla^2\\psi}\\bigg)\n        +\\rho\\frac{\\partial \\mathfrak{u}}{\\partial \\nabla^2\\psi}\\mathbf{D}\\psi\n    \\bigg\\} : \\nabla \\mathbf{v} \\\\[2mm]\n    && +\\bigg[ \\boldsymbol\\xi -\\rho \\frac{\\partial \\mathfrak{u}}{\\partial\\nabla\\psi} + \\nabla\\left(\\rho\\frac{\\partial \\mathfrak{u}}{\\partial \\nabla^2 \\psi}\\right\n        )\\bigg] \\cdot \\nabla \\dot{\\psi} + \\bigg[\\kappa\\zeta\\Big(\\rho-\\rho_0-\\kappa A\\Big)\\frac{\\partial A}{\\partial\\psi}-\\rho\\frac{\\partial \\mathfrak{u}}{\\partial \\psi}+\\nabla\\cdot\\boldsymbol\\xi\\bigg]\\dot{\\psi} \\; .\n\\label{eq:sbal-w}\n\\end{eqnarray}\n\n\\vspace{5mm}\n\nIn equilibrium, we observe from Eq.~(\\ref{eq:sbal-w}) that the balance of microforces is satisfied for\n\\begin{eqnarray}\n  \\boldsymbol\\xi &=& \\rho \\frac{\\partial \\mathfrak{u}}{\\partial\\nabla\\psi} - \\nabla\\left(\\rho\\frac{\\partial \\mathfrak{u}}{\\partial \\nabla^2 \\psi}\\right) \\;,\n  \\\\[2mm]\n  \\pi&=& \\kappa\\zeta\\Big(\\rho-\\rho_0-\\kappa A\\Big)\\frac{\\partial A}{\\partial\\psi}-\\rho\\frac{\\partial \\mathfrak{u}}{\\partial \\psi} \\;.\n\\end{eqnarray}\nThe terms in square brackets proportional to $\\dot{\\psi}$ and $\\nabla\\dot{\\psi}$ in Eq.~(\\ref{eq:sbal-w}) are both related to variations of $\\psi$ and can be grouped together. Accounting for the boundary conditions, we obtain the final form of the local entropy balance\n\\begin{eqnarray}\n    \\nonumber\n    \\rho \\theta \\dot{\\mathfrak{s}} &=& \n   \n     \\bigg\\{ \\mathbf{T} + \\frac{\\zeta}{2}\\Big[\\rho^2-(\\rho_0+\\kappa A)^2 \\Big]\\mathbf{I}\n        +\\rho \\nabla\\psi \\otimes \\frac{\\partial \\mathfrak{u}}{\\partial\\nabla\\psi}\n        -\\nabla\\psi\\otimes\\nabla\n        \\bigg(\\rho\\frac{\\partial \\mathfrak{u}}{\\partial \\nabla^2\\psi}\\bigg)\n        +\\rho\\frac{\\partial \\mathfrak{u}}{\\partial \\nabla^2\\psi}\\mathbf{D}\\psi\n    \\bigg\\} : \\nabla \\mathbf{v} \\\\[2mm]\n    && +\\bigg[\\kappa\\zeta\\Big(\\rho-\\rho_0-\\kappa A\\Big)\\frac{\\partial A}{\\partial\\psi}-\\rho\\frac{\\partial \\mathfrak{u}}{\\partial \\psi}\n    +\\nabla\\cdot\\bigg(\\rho \\frac{\\partial \\mathfrak{u}}{\\partial\\nabla\\psi}\\bigg)\n    -\\nabla^2\\bigg(\\rho\\frac{\\partial \\mathfrak{u}}{\\partial \\nabla^2 \\psi}\\bigg) \\bigg] \\dot{\\psi}\n    \\; .\n\\label{eq:sbal2-w}\n\\end{eqnarray}\nNote that the microstress is not explicit in this form of the entropy balance.\n\n\\subsection{Governing equations}\n\nConstitutive relations and governing equations are now derived by imposing strict requirements on the entropy production $\\dot{\\mathfrak{s}}$, as in the Coleman-Noll procedure~\\cite{coleman1963thermodynamics,hutter2020coleman}. The Clausius-Duhem inequality states that every admissible thermomechanical process must satisfy $\\dot{\\mathfrak{s}} \\geq 0$, which has direct implications for Eq.~(\\ref{eq:sbal2-w}). We start by splitting the stress into a sum of reversible and dissipative parts, $\\mathbf{T} = \\mathbf{T}^R + \\mathbf{T}^D$, so that the reversible part $\\mathbf{T}^R$ can be derived from Eq.~(\\ref{eq:sbal2-w}) in the limit of zero entropy production, while dissipative parts are obtained by enforcing positive entropy production.\n\nThe reversible stress $\\mathbf{T}^R$, is obtained from the expression inside the brackets associated with the rate $\\nabla\\mathbf{v}$ in the limit $\\dot{\\mathfrak{s}} = 0$. We find\n\\begin{eqnarray}\n    \\mathbf{T}^R &=& -\\frac{\\zeta}{2}\\Big[\\rho^2-(\\rho_0+\\kappa A)^2 \\Big]\\mathbf{I}\n        - \\rho \\nabla\\psi \\otimes \\frac{\\partial \\mathfrak{u}}{\\partial\\nabla\\psi}\n\n             +\\nabla\\psi\\otimes\\nabla\n             \\bigg(\\rho\\frac{\\partial \\mathfrak{u}}{\\partial \\nabla^2\\psi}\\bigg)\n             -\\rho\\frac{\\partial \\mathfrak{u}}{\\partial \\nabla^2\\psi}\\mathbf{D}\\psi \\; .\n        \\label{eq:rev-stress-w}\n\\end{eqnarray}\nNote that non-classical stress terms appear in Eq.~(\\ref{eq:rev-stress-w}), not only from the dependence of $\\mathfrak{u}$ on $\\nabla^2\\psi$, but also from the density coupling through $\\zeta$. Generally, we will expect the latter to be small for the bulk smectic and isotropic phases, since $\\rho$ tends to approach $\\rho_0 + \\kappa A$ in the bulk due to energy minimization. However, this term can potentially become large for compressible flows near the smectic-isotropic interface.\n\nThe generalized chemical potential $\\mu$, which is the the thermodynamic conjugate to $\\psi$, $\\mu = \\delta \\mathfrak{U} \/ \\delta \\psi$, appears in Eq.~(\\ref{eq:rev-stress-w}) inside the brackets multiplying $\\dot{\\psi}$. That is,\n\\begin{eqnarray}\n  \\mu &=&\n          - \\kappa\\zeta\\Big(\\rho-\\rho_0-\\kappa A\\Big)\\frac{\\partial A}{\\partial\\psi}\n          + \\rho\\frac{\\partial \\mathfrak{u}}{\\partial \\psi}\n    -\\nabla\\cdot\\bigg(\\rho \\frac{\\partial \\mathfrak{u}}{\\partial\\nabla\\psi}\\bigg)\n    +\\nabla^2\\bigg(\\rho\\frac{\\partial \\mathfrak{u}}{\\partial \\nabla^2 \\psi}\\bigg) \\; .\n    \\label{eq:chemicalp-w}\n\\end{eqnarray}\nFor a slowly relaxing variable such as $\\psi$, which is not a conserved quantity, the associated dynamic equation~\\cite{pleiner1996hydrodynamics} is given by a quasi-current $Z$, and takes the form \n\\begin{equation}\n    \\partial_t\\psi + \\mathbf{v}\\cdot\\nabla\\psi + Z = 0\\;.\n    \\label{eq:psydyn}\n\\end{equation}\nWhile reversible motion requires $\\dot{\\mathfrak{s}} = 0$ in Eq.~(\\ref{eq:sbal2-w}), the generalized chemical potential $\\mu$ from Eq.~(\\ref{eq:chemicalp-w}) is arbitrary, and reversible motion has $\\dot{\\psi} = 0$. The implication is that $Z$ has no reversible part, and is purely dissipative.\n\nIrreversible currents are obtained by requiring a positive entropy production $\\dot{\\mathfrak{s}} \\geq 0$. From Eq.~(\\ref{eq:sbal2-w}), $\\dot{\\psi}$ must be proportional to the chemical potential $\\mu$ times a mobility constant $\\Gamma$, so that\n\\begin{equation}\n    Z = \\Gamma\\mu \\;.\n\\end{equation}\nThe irreversible stress $\\mathbf{T}^D$ when contracted with $\\nabla\\mathbf{v}$ in Eq.~\\ref{eq:sbal2-w} should result in a positive quantity to satisfy the Clausius-Duhem inequality. This implies that $\\mathbf{T}^D = \\boldsymbol\\eta:\\nabla\\mathbf{v}$, where $\\boldsymbol\\eta$ is a fourth order viscosity tensor. For simplicity, we will restrict our analysis for a Newtonian fluid, with a dissipative stress of the form\n\\begin{equation}\n    \\mathbf{T}^D = \\eta(\\nabla\\mathbf{v}+\\nabla\\mathbf{v}^\\top)+\\lambda(\\nabla\\cdot\\mathbf{v})\\mathbf{I}\\;,\n    \\label{eq:dstress}\n\\end{equation}\nwhere $\\eta$ and $\\lambda$ are the first and second coefficient of viscosity, respectively. The coefficient $\\lambda$ is important for compressibility effects, as it controls the magnitude of the longitudinal part of the flow. We note that the dissipative stress for a general uniaxial phase can be written in terms of five independent viscosity coefficients~\\cite{degennes1995physics}, as we argued in~\\cite{vitral2020model}. While an extension to such a dissipative stress is straightforward in the model, these coefficients are poorly characterized experimentally, so that we focus exclusively on the simpler form of Eq.~(\\ref{eq:dstress}). Further, we consider $\\eta$ and $\\lambda$ to be homogeneous throughout the smectic-isotropic system, so that another possible extension of the model would consider viscosity contrast between the phases.\n\nFrom the constitutive relations for the stress $\\mathbf{T}$ and the quasi-current $Z$, we find that the balance of mass, the balance of linear momentum, and the dynamic equation for the order parameter, which govern the weakly compressible smectic-isotropic system, have the following form\n\\begin{eqnarray}\n    \\dot{\\rho} &=& \n    -\\rho\\nabla\\cdot\\mathbf{v} \\; ,\n    \\label{eq:wcom-bm}\n    \\\\[2mm]\n    \\rho\\dot{\\mathbf{v}} &=&\n    \\nabla\\cdot\\Big(\\mathbf{T}^R + \\mathbf{T}^D\\Big) \\; ,\n    \\label{eq:wcom-blm}\n    \\\\[2mm]\n    \\dot{\\psi} &=& -\\Gamma \\mu \\;.\n    \\label{eq:wcom-psi}\n\\end{eqnarray}\nBoundary conditions are specified by Eqs.~(\\ref{eq:bc-w}), the reversible stress $\\mathbf{T}^{R}$ is defined in Eq.~(\\ref{eq:rev-stress-w}), the dissipative stress $\\mathbf{T}^{D}$ in Eq.~(\\ref{eq:dstress}) and the generalized chemical potential $\\mu$ in Eq.~(\\ref{eq:chemicalp-w}).\n\n\\subsection{Energy density of the smectic phase}\n\nThe specific energy $\\mathfrak{u}$ we use allows for coexistence of isotropic and smectic phases, and remains rotationally invariant. It is given by\n\\begin{eqnarray}\n    \\mathfrak{u}(\\psi,\\nabla^2\\psi) &=& \\frac{1}{2}\\bigg\\{ \\epsilon \\psi^2 \n    + \\alpha\\left[ \\left(\\nabla^2+q_0^2 \\right)\\psi \\right]^2            \n    - \\frac{\\beta}{2} \\psi^{4} + \\frac{\\gamma}{3} \\psi^{6}\\;\\bigg\\},\n    \\label{eq:energy-density}\n\\end{eqnarray}\nwhere $\\psi = 0$ represents the isotropic phase, and a periodic $\\psi$ the smectic phase. The coefficients $\\alpha$, $\\beta$ and $\\gamma$ are three constant, positive parameters, $\\epsilon$ is a bifurcation parameter that describes the distance away from the smectic-isotropic transition, and $q_{0}$ is approximately the smectic layering wave number. Hence, this energy penalizes deviations from the preferred equidistant layering~\\cite{napoli1999smectic}. The associated quintic Swift-Hohenberg equation has been used in the past to describe modulated phases~\\cite{sakaguchi1996stable,sakaguchi1998localized,burke2007snakes} in coexistence with an isotropic ($\\psi=0$) phase, and describes the diffusive relaxation of the order parameter $\\psi$ driven by energy minimization. The sixth order polynomial in $\\psi$ leads to a triple well energy potential function, whose minima represent the smectic and isotropic phases. By fixing $\\beta$ and $\\gamma$, the relative height between these wells can be controlled through $\\epsilon$, such that smectric-isotropic coexistence occurs at $\\epsilon_{c} = 27 \\beta^{2}\/160 \\gamma$, when both phases present the same energy density. For $\\epsilon > \\epsilon_{c}$, $\\psi = 0$ (isotropic) becomes the equilibrium phase, whereas for $\\epsilon < \\epsilon_{c}$, a modulated phase (smectic) $\\psi \\approx \\frac{1}{2} (A\\,e^{i{\\bf q}\\cdot{\\bf x}} + c.c.)$ is the equilibrium one. The amplitude $A_s$ of the bulk smectic is given by~\\cite{sakaguchi1996stable}\n\\begin{equation}\n    A_s^2 = \\frac{3\\beta+\\sqrt{9\\beta^2-40\\epsilon\\gamma}}{5\\gamma} \\;,\n\\end{equation} \nwhich is relevant when choosing the constants $\\kappa$ and $\\rho_0$ for a specific density ratio. For small layer perturbations away from planarity, the energy reduces to the classical Oseen-Frank form of the smectic energy, where the elastic moduli for compression of layers and splay of molecules can be written as a function of $\\alpha$~\\cite{vitral2019role}.\n\nBy substituting this definition of specific energy into the chemical potential from Eq.~(\\ref{eq:chemicalp-w}), we obtain\n\\begin{eqnarray}\n\n    \\mu &=&  - \\kappa\\zeta\\Big(\\rho-\\rho_0-\\kappa A\\Big)\\frac{\\partial A}{\\partial\\psi}+\\rho\\Big[\\epsilon\\psi +\\alpha q_0^2(\\nabla^2+q_0^2)\\psi-\\beta\\psi^3+\\gamma\\psi^5\\Big] \n    + \\alpha\\nabla^2\\Big[\\rho(\\nabla^2+q_0^2)\\psi\\Big]\\;.\n    \\label{eq:chemicalp2-w}\n\\end{eqnarray}\n\nAn issue concerning the actual computation of $\\mu$ from Eq.~(\\ref{eq:chemicalp2-w}) is that while we are able to numerically extract the amplitude from $\\psi$, the dependence of $A$ as a function of $\\psi$ is unknown, and so is its derivative with respect to $\\psi$. In this work, since the order parameter $\\psi$ has an asymptotic sinusoidal form, we compute the amplitude $A$ through $A = (\\psi^2+q_0^{-2}|\\nabla\\psi|^2)^{1\/2}$. By accounting for this dependency on $\\nabla\\psi$, we then write the chemical potential $\\mu = \\delta \\mathfrak{U} \/ \\delta \\psi$ as,\n\\begin{eqnarray}\n  \\nonumber\n  \\mu &=&  - \\kappa\\zeta\\Big(\\rho-\\rho_0-\\kappa A\\Big)\\frac{\\psi}{A}+\\kappa\\zeta q_0^{-2}\\nabla\\cdot\\bigg[(\\rho-\\rho_0-\\kappa A)\\frac{\\nabla\\psi}{A} \\bigg]\n  \\\\[2mm]&&\n            +\\rho\\Big[\\epsilon\\psi +\\alpha q_0^2(\\nabla^2+q_0^2)\\psi-\\beta\\psi^3+\\gamma\\psi^5\\Big] \n    + \\alpha\\nabla^2\\Big[\\rho(\\nabla^2+q_0^2)\\psi\\Big]\\;.\n    \\label{eq:chemicalp3-w}\n\\end{eqnarray}\n\nWe also neglect inertia compared to viscous effects, so that by using the definition of the energy density in Eq.~(\\ref{eq:energy-density}), the balance of linear momentum, Eq.~(\\ref{eq:wcom-blm}), becomes\n\\begin{eqnarray}\n    \\nonumber\n  \\mathbf{0} &=&  -\\frac{\\zeta}{2}\\nabla\\Big[\\rho^2-(\\rho_0+\\kappa A)^2 \\Big]\n        + \\alpha\\nabla^2\\Big[\\rho(\\nabla^2+q_0^2)\\psi\\Big]\\nabla\\psi        \n  \\\\[2mm]\n    &&\n       -\\alpha\\rho(\\nabla^2+q_0^2)\\psi\\nabla^2\\nabla\\psi\n       + \\eta \\nabla^2\\mathbf{v} \n       + (\\eta + \\lambda)\\nabla(\\nabla\\cdot\\mathbf{v}) \\; .\n    \\label{eq:wcom-stokes}\n\\end{eqnarray}\nBy defining the modified chemical potential $\\bar{\\mu}$ as\n\\begin{eqnarray}\n    \\nonumber\n  \\bar{\\mu} &=&  \n    \\rho\\Big[\\epsilon\\psi +\\alpha q_0^2(\\nabla^2+q_0^2)\\psi-\\beta\\psi^3+\\gamma\\psi^5\\Big] \n    + \\alpha\\nabla^2\\Big[\\rho(\\nabla^2+q_0^2)\\psi\\Big]\\;,\n    \\label{eq:chemicalp4-w}\n\\end{eqnarray}\nwe can rewrite Eq.~(\\ref{eq:wcom-stokes}) as,\n\\begin{eqnarray}\n  \\mathbf{0} &=&  -\\frac{\\zeta}{2}\\nabla\\Big[\\rho^2-(\\rho_0+\\kappa A)^2 \\Big]\n        + \\bar{\\mu}\\nabla\\psi - \\rho\\nabla\\mathfrak{u}\n       + \\eta \\nabla^2\\mathbf{v} \n       + (\\eta + \\lambda)\\nabla(\\nabla\\cdot\\mathbf{v}) \\; .\n    \\label{eq:wcom-stokes2}\n\\end{eqnarray}\nThe quantity $\\bar{\\mu}\\nabla\\psi$ is known as the osmotic force \\cite{fielding2003flow}, and is exactly the forcing term that appears for an incompressible smectic-isotropic system, where both phases have the same density~\\cite{vitral2019role}. Therefore, $-\\rho\\nabla \\mathfrak{u}$ is a force that originates from compressibility effects. \n\nDimensionless variables are introduced similarly to~\\cite{vitral2020model}. Let $U$ and $L$ represent characteristic scales for the velocity and length, and $\\tilde{\\rho}$ and $\\tilde{\\mu}$ represent typical values for $\\rho$ and $\\mu$ in the modulated phase. We then perform the non-dimensionalization by defining $\\mathbf{v}^* = \\mathbf{v}\/U$, $\\mathbf{x}^* = \\mathbf{x}\/L$, $t^* = U t\/L$,  $\\rho^* = \\rho\/\\tilde{\\rho}$, $\\mu^* = \\mu\/\\tilde{\\mu}$ and $\\psi^* = \\psi\\tilde{\\mu}\/\\tilde{\\rho}U^2$. The dimensionless constants one finds are $\\kappa^* = \\kappa \/ \\tilde{\\rho}$, $\\Gamma^* = \\Gamma \\tilde{\\mu}^2 L \/ \\tilde{\\rho} U^3$, $\\zeta^* = \\zeta \\tilde{\\rho}\/U^2$, $\\eta^* = \\eta \/ \\tilde{\\rho}U L$ and $\\lambda^* = \\lambda \/ \\tilde{\\rho}U L$. Note that $\\eta^*$ is the inverse Reynolds number, and $\\zeta^*$ is a dimensionless number that controls the ratio of $\\psi$ currents arising from density gradients compared to $\\psi$ owing to local curvatures. \nThis can be seen by taking the ratio of terms in Eq.~(\\ref{eq:chemicalp2-w}) in dimensionless form:\n\\begin{equation}\n    \\frac{\\kappa\\zeta\\delta\\rho}{\\bar{\\mu}}\\,\\frac{\\partial A}{\\partial \\psi}\n    = \\zeta^* \\bigg(\\frac{\\kappa^*\\delta\\rho^*}{\\bar{\\mu}^*}\\,\\frac{\\partial A}{\\partial \\psi^*}\\bigg) \\;,\n\\end{equation}\nwhere $\\delta\\rho = \\rho-\\rho_{eq}$ and the chemical potential $\\bar{\\mu}$ is proportional to local curvatures through the Gibbs-Thomson relation, see Eqs.~(\\ref{eq:gibbs-thomson}) and (\\ref{eq:gt-perp}). Hence, when $\\zeta$ is small, the evolution of $\\psi$ is driven primarily by the chemical potential $\\bar{\\mu}$.\n\nBy dropping the star notation from variables and constants, we here summarize the complete set of dimensionless governing equations for the weakly compressible smectic-isotropic fluid system, including balance of mass, balance of linear momentum, and order parameter equation,\n\\begin{eqnarray}\n  \\label{eq:bm-ge-w}\n  \\dot{\\rho}\n  &=& -\\rho\\nabla\\cdot\\mathbf{v}\\; ,\n  \\\\[4mm]\n   \\label{eq:blm-ge-w}\n  \\mathbf{0}\n  &=& -\\frac{\\zeta}{2}\\nabla\\Big[\\rho^2-(\\rho_0+\\kappa A)^2 \\Big]\n      +\\bar{\\mu}\\nabla\\psi -\\rho\\nabla \\mathfrak{u} \n      + \\eta \\nabla^2\\mathbf{v} \n      + (\\eta + \\lambda)\\nabla(\\nabla\\cdot\\mathbf{v}) \\; ,\n  \\\\[4mm]\n  \\label{eq:psi-ge-w}\n  \\dot{\\psi} &=& -\\Gamma \\mu \\; ,\n\\end{eqnarray}\nwith the generalized chemical potential $\\mu$ given by  Eq.~(\\ref{eq:chemicalp3-w}), modified chemical potential $\\bar{\\mu}$ given by Eq.~(\\ref{eq:chemicalp4-w}), and energy density $\\mathfrak{u}$ given by Eq.~(\\ref{eq:energy-density}). Note that $\\zeta$ in Eq.~(\\ref{eq:blm-ge-w}) plays a role in setting diffuseness of the density across the interface.  For small $\\zeta$ the interface is more diffuse and the penalty for deviations of $\\rho$ from equilibrium is small, while for larger $\\zeta$ the variation in $\\rho$ across the interface becomes sharper and the system approaches quasi-incompressibility.\n\n\\section{Numerical algorithm}\n\\label{sec:ns}\n\nThe governing equations~(\\ref{eq:bm-ge-w})-(\\ref{eq:psi-ge-w}), with boundary conditions specified in Eq.~(\\ref{eq:bc-w}), are integrated with a pseudo-spectral method in three spatial dimensions, in which linear and gradient terms are computed in Fourier space and nonlinear terms in real space. A regular cubic grid is used with linear spacing $\\Delta x = 2\\pi\/(n_w q_0)$, where $n_w$ is the number of grid points per base wavelength. We have developed a custom C++ code (\\textit{smaiso-wcomp}) based on the parallel FFTW library and the standard MPI passing interface for parallelization, which is publicly avilable~\\cite{smaiso-wcomp}. In order to accommodate the boundary conditions, we use both the Discrete Cosine Transform of ($\\psi$, $\\rho$) and the Discrete Sine Transform of ($\\nabla\\psi$, $\\mathbf{v}$).\n\nWe compute the order parameter amplitude $A$ by $A = (\\psi^2+q_0^{-2} |\\nabla\\psi|^2)^{1\/2}$.  While this approximation gives us an adequate value of $A$ in regions where the smectic layers are well formed and only weakly distorted, it becomes noisier on the interface, and also in regions where layers are highly distorted or break up. Therefore in our numerical calculations we smooth the computed amplitude with a Gaussian filter in Fourier space, given by the operator $F_{\\omega} = \\textrm{exp}(-\\omega^2 q^2\/2)$, where $q$ is the local wavenumber and $\\omega$ the filtering radius, chosen as $1\/q_0$.\n\n\\subsection{Order parameter equation}\n\nThe numerical scheme for integrating Eq.~(\\ref{eq:psi-ge-w}) has been detailed in~\\cite{vitral2020model}, and here we summarize it. Due to the variable density multiplying the RHS of Eq.~(\\ref{eq:psi-ge-w}), it cannot be dealt with in the same form as the uniform density case~\\cite{vitral2019role}, for which the split in linear and nonlinear parts was immediate. Instead, we follow a scheme previously employed for phase-field models with variable mobility~\\cite{zhu1999coarsening,badalassi2003computation}. First, we split Eq.~(\\ref{eq:psi-ge-w}) as $\\partial_t\\psi = \\Gamma(\\rho L \\psi + N)$ with\n\\begin{eqnarray}\n    L &=& -\\Big[\\epsilon + (\\nabla^2+q_0^2)^2\\Big] \\;,\n    \\label{eq:linear-w}\n    \\\\[2mm]\n    \\nonumber\n    N &=&  \\kappa\\zeta\\Big(\\rho-\\rho_0-\\kappa A\\Big)\\frac{\\psi}{A}-\\kappa\\zeta q_0^{-2}\\nabla\\cdot\\bigg[(\\rho-\\rho_0-\\kappa A)\\frac{\\nabla\\psi}{A}\\bigg]\n    \\\\[2mm] &&\n    -2\\alpha\\nabla\\rho\\cdot(\\nabla^2+q_0^2)\\nabla\\psi\n    -\\alpha\\nabla^2\\rho\\,(\\nabla^2+q_0^2)\\psi+\\beta\\psi^3-\\gamma\\psi^5\n    -\\Gamma^{-1}\\mathbf{v}\\cdot\\nabla\\psi  \\;,\n    \\label{eq:nlinear-w}\n\\end{eqnarray}\nwhere $L$ is a linear operator, and $N$ is a collection of nonlinear terms. We split the density as $\\rho \\rightarrow \\rho_m +(\\rho-\\rho_m)$, where $\\rho_m = \\frac{1}{2}(\\rho_s + \\rho_0)$. Here, $\\rho_s$ is the density of the smectic bulk, which can be obtained from the system parameters by $\\rho_s = \\kappa A_s + \\rho_0$, where $A_s$ is the amplitude solution of the sinusoidal phase. The term associated with $\\rho_m$ can be treated implicitly, and $(\\rho-\\rho_m)$ is treated explicitly, with a choice of $\\rho_m$ that satisfies $|\\rho -\\rho_m| \\leq \\rho_m$. We treat the linear term $L$ implicitly, while the nonlinear term $N$ is treated with a second order Adams-Bashforth scheme. In Fourier space, the order parameter $\\psi_q$ for the new time step $n+1$ is computed from\n\\begin{eqnarray}\n    (3\/2-\\Delta t\\, \\Gamma\\rho_m L )\\psi_q^{n+1}\n    \\,=\\,  (2-\\Delta t\\, \\Gamma\\rho_m L)\\psi_q^{n}-\\frac{1}{2}\\psi_q^{n-1}\n    +\\frac{\\Delta t\\, \\Gamma}{2}(3N_q^n-N_q^{n-1}) \\; .\n    \\label{eq:method2-w}\n\\end{eqnarray}\nNumerical verification and stability of this scheme have been studied in~\\cite{vitral2020model}, where we found that for $n_w = 8$ a time step of $\\Delta t = 1\\cdot 10^{-3}$ or less was necessary to guarantee stability.\n\n\\subsection{Velocity decomposition}\n\nThe weakly compressible model requires additional considerations for computing the velocity and density as compared to~\\cite{vitral2020model}. A Helmholtz decomposition the velocity field $\\mathbf{v}$ is introduced,\n\\begin{eqnarray}\n  \\mathbf{v} &=& \\nabla\\Phi + \\nabla\\times\\mathbf{P} \\;,\n  \\label{eq:helmholtz}\n\\end{eqnarray}\nwhere $\\Phi$ is a scalar potential and $\\mathbf{P}$ is a vector potential. By substituting this decomposition of the velocity into the balance of linear momentum from Eq.~(\\ref{eq:blm-ge-w}), we obtain\n\\begin{eqnarray}\n  \\mathbf{0} &=&  -\\frac{\\zeta}{2}\\nabla\\Big[\\rho^2-(\\rho_0+\\kappa A)^2 \\Big]\n       + \\mathbf{f}   \n       + \\eta \\nabla^2\\nabla\\times\\mathbf{P} \n       + (2\\eta + \\lambda)\\nabla\\nabla^2\\Phi \\; ,\n       \\label{eq:blm-helm}\n\\end{eqnarray}\nwhere $\\mathbf{f} = \\bar{\\mu}\\nabla\\psi-\\rho\\nabla \\mathfrak{u}$. Since the gradient of the density is computed numerically at every time step, one can avoid computing the gradient of the energy density $\\mathfrak{u}$ by adding $\\rho \\mathfrak{u}$ inside the gradient from the first term in Eq.~(\\ref{eq:blm-helm}) and rewriting $\\mathbf{f}$ as $\\bar{\\mathbf{f}} = \\bar{\\mu}\\nabla\\psi + \\mathfrak{u}\\nabla \\rho$.\n\nThe solenoidal field can be obtained from the transverse part of Eq.~(\\ref{eq:blm-helm}). By eliminating irrotational terms through an orthogonal projection (and, consequently, modulations due to the layering), we compute $\\nabla\\times\\mathbf{P}$ in Fourier space from\n\\begin{eqnarray}\n     (\\nabla\\times\\mathbf{P})_q\n     &=& \\frac{1}{\\eta\\, q^2}\\bigg(\\mathbf{I} - \n     \\frac{\\mathbf{q}\\otimes\\mathbf{q}}{q^2} \\bigg)\\mathbf{f}_q \\;.\n     \\label{eq:transv}\n\\end{eqnarray}\nSimilarly, the longitudinal component of Eq.~(\\ref{eq:blm-helm}) eliminates the solenoidal terms, and allows us to compute $\\nabla\\Phi$. By substituting Eq.~(\\ref{eq:transv}) into Eq.~(\\ref{eq:blm-helm}), we obtain \n\\begin{eqnarray}\n     (\\nabla \\Phi)_q\n     &=& \\frac{1}{(2\\eta +\\lambda)\\, q^2}\\bigg\\{\\frac{\\zeta\\,\\mathbf{q}}{2}\n     \\Big[\\rho^2-(\\rho_0+\\kappa A)^2 \\Big]_q + \n     \\frac{\\mathbf{q}\\otimes\\mathbf{q}}{q^2}\\,\\mathbf{f}_q \\bigg\\} \\;.\n     \\label{eq:irrot}\n\\end{eqnarray}\nNote that the density does not change in the scale of the smectic layering modulations, and also that the force $\\mathbf{f}$ contains both resonant terms on the same scale as the amplitude, and modes $\\pm 2 i q_0$ or higher. In order to avoid spurious oscillations in the irrotational flow along smectic layers, we dampen contributions from the higher order frequencies by applying the filter $F_{\\omega} = \\textrm{exp}(-\\omega^2 q^2\/2)$ to the $\\mathbf{f}_q$ term in Eq.~(\\ref{eq:irrot}), with a filtering radius $\\omega = 1\/q_0$.\n\n\\subsection{Balance of mass}\n\nThe balance of mass from Eq.~(\\ref{eq:bm-ge-w}) can be easily integrated by a multistep method such as Adam-Bashforth. Here, we compute the density at the new time step $n+1$ through\n\\begin{eqnarray}\n    \\rho^{n+1} &=& \\rho^n -\n    \\Delta t \\bigg[\n    \\frac{3}{2}\\big(\\mathbf{v}^n\\cdot\\nabla\\rho^n\n    +\\rho^n \\nabla\\cdot\\mathbf{v}^n\\big)\n \n             -\\frac{1}{2}\\big(\\mathbf{v}^{n-1}\\cdot\\nabla\\rho^{n-1}\n             +\\rho^{n-1} \\nabla\\cdot\\mathbf{v}^{n-1}\\big) \\bigg] \\;,\n\\end{eqnarray}\nwhere the divergence of the velocity is computed from $\\nabla\\cdot\\mathbf{v} = \\nabla^2\\Phi$.\n\n\\section{Compressibility effects on order parameter diffusion}\n\\label{sec:def}\n\nOne of the main goals of the weakly compressible model is to explore how the energy penalty for  deviations in density from equilibrium in Eq.~(\\ref{eq:energy}) affects the evolution of the non-conserved order parameter $\\psi$.  We have shown that the dimensionless $\\zeta$ controls the ratio between order parameter currents arising from density gradients and curvatures. Now we investigate its role on the motion of $\\psi$ given by Eq.~(\\ref{eq:psi-ge-w}).\n\nAssume first we have a region of constant density and amplitude $A$. By setting $\\Gamma = 1$, as in our numerical computations, the order parameter equation~(\\ref{eq:psi-ge-w}) can be written as\n\\begin{eqnarray}\n  \\dot{\\psi} &=&  \\frac{\\kappa\\zeta}{A}\\Big(\\rho-\\rho_0-\\kappa A\\Big)(1-q_0^{-2}\\nabla^2)\\psi\n            -\\rho\\Big[\\epsilon\\psi +\\alpha(\\nabla^2+q_0^2)^2\\psi-\\beta\\psi^3+\\gamma\\psi^5\\Big] \n   \\;.\n    \\label{eq:psi-const}\n\\end{eqnarray}\nThe first term on the RHS is linear under this scenario. Since the order parameter asymptotic solution is \\hbox{$\\psi \\approx A \\,\\textrm{cos}(\\mathbf{q}_0\\cdot\\mathbf{x})$}, this implies that $(1-q_0^{-2}\\nabla^2)\\psi \\approx 2\\psi$. By using this approximation  we can regroup the terms in Eq.~(\\ref{eq:psi-const}) as\n\\begin{eqnarray}\n\\nonumber\n  \\dot{\\psi} &\\approx&  -[\\rho\\epsilon-2\\frac{\\kappa\\zeta}{A}(\\rho - \\rho_0 -\\kappa A)]\\psi\n            -\\rho\\Big[\\alpha(\\nabla^2+q_0^2)^2\\psi-\\beta\\psi^3+\\gamma\\psi^5\\Big]\n   \\\\[2mm]\n   &=& -\\rho\\epsilon_{e}\\psi\n   -\\rho\\Big[\\alpha(\\nabla^2+q_0^2)^2\\psi-\\beta\\psi^3+\\gamma\\psi^5\\Big] \\;.\n    \\label{eq:psi-const2}\n\\end{eqnarray}\nTherefore, the term proportional to $\\zeta$ can be viewed as a correction to the bifurcation parameter $\\epsilon$, resulting in an effective bifurcation parameter $\\epsilon_e$.\n\nWe first investigate the stability of our algorithm by computing a stationary configuration at coexistence with planar smectic-isotropic interfaces, as well as the evolution near coexistence to evaluate numerically the effect of $\\epsilon_e$.  The governing equations are integrated with a time step $\\Delta t = 0.001$ and grid spacing $\\Delta x = \\pi\/4$ (8 points per wavelength). These equilibrium configurations have been obtained by setting  $q_0 = 1$, $\\beta = 2$, $\\gamma = 1$, $\\nu = 1$, $\\lambda = 1$, $\\zeta = 1$, an isotropic equilibrium density $\\rho_0 = 0.5$ and $\\kappa = 0.3727$, so that the equilibrium density of the smectic is $\\rho_s = 1$ (density ratio $\\rho_s:\\rho_0 = 2:1$). \n\nConsider a stack of planar smectic layers in contact with an isotropic phase, where the layer normal is in the $\\hat{z}$ direction. When both phases are at coexistence ($\\epsilon = \\epsilon_c$), the density profile is $\\rho = \\rho_0 + \\kappa A$, as illustrated in Fig.~\\ref{fig:flat-ic}. This figure shows the $xz$ cross-section of a cubic computational domain with $N = 64^3$ nodes at coexistence with $\\epsilon_c = 0.675$.  The velocity is $\\mathbf{v} = \\mathbf{0}$, and the interface is stationary with $\\partial_t\\psi = 0$. If $\\epsilon \\neq \\epsilon_{c}$ there is a direct coupling between the local density difference relative to the stationary case and the order parameter. This coupling term generally opposes interfacial motion.\n\n\\begin{figure}[ht]\n\t\\centering\n    \\begin{subfigure}[b]{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/psi-rho2to1-t0.pdf}\n    \\caption{$\\psi$}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/rho-rho2to1-t0.pdf}\n    \\caption{$\\rho$}\n    \\end{subfigure}\n    \\caption{Middle cross section $xy$ of a stack of flat smectic layers with normal in the $z$ direction, showing the order parameter field $\\psi$ and density field $\\rho$ used as initial condition. Simulations employing this initial conditions use parameters $\\beta = 2$, $\\gamma = 1$, $\\nu = 1$, $\\lambda = 1$, $\\rho_0 = 0.5$ and $\\kappa = 0.3727$. For these values, the coexistence parameter is $\\epsilon_c = 0.675$.}\n\t\\label{fig:flat-ic}\n\\end{figure}\n\nStarting from this initial configuration, Fig.~\\ref{fig:flat-param-a} shows the evolving configuration at time $t = 90$ when $\\epsilon$ is set to $0.5$ (within the smectic region of the phase diagram), and coupling constant $\\zeta =1$. The initial smectic has grown, and the region of high density has also spread, with its value reduced from the initial $\\rho_s = 1$ (so as to conserve mass). We note that the smectic is only able to grow significantly due to the fact that $\\rho_s:\\rho_0 = 2:1$ so that there are enough molecules in the isotropic phase to allow growth.  Smectic growth was not observed for larger density ratios such as $\\rho_s:\\rho_0 = 100:1$ when $\\zeta = 1$. \n\n\\begin{figure}[ht!]\n\t\\centering\n    \\begin{subfigure}[b]{0.24\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/psi-rho2to1-e0d5-t90.pdf}\n        \\includegraphics[width=\\textwidth]{figures\/rho-rho2to1-e0d5-t90.pdf}\n        \\caption{$\\epsilon = 0.5,\\,\\zeta = 1$}\n        \\label{fig:flat-param-a}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.24\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/psi-rho2to1-z100-e0d5-t90.pdf}\n        \\includegraphics[width=\\textwidth]{figures\/rho-rho2to1-z100-e0d5-t90.pdf}\n        \\caption{$\\epsilon = 0.5,\\,\\zeta = 100$}\n        \\label{fig:flat-param-b}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.24\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/psi-rho2to1-e0d8-t90.pdf}\n        \\includegraphics[width=\\textwidth]{figures\/rho-rho2to1-e0d8-t90.pdf}\n        \\caption{$\\epsilon = 0.8,\\,\\zeta = 1$}\n        \\label{fig:flat-param-c}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.24\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/psi-rho2to1-z100-e0d8-t90.pdf}\n        \\includegraphics[width=\\textwidth]{figures\/rho-rho2to1-z100-e0d8-t90.pdf}\n        \\caption{$\\epsilon = 0.8,\\,\\zeta = 100$}\n        \\label{fig:flat-param-d}\n    \\end{subfigure}\n    \\caption{Order parameter (top) and density (bottom) configuration at time $t = 90$, starting from the initial condition in Fig. \\ref{fig:flat-ic}. For $\\epsilon = 0.5$, the smectic phase is energetically favored, while for $\\epsilon  = 0.8$, the isotropic phase is energetically favored. For $\\zeta = 1$ the interface moves to grow the energetically favored phase, while for $\\zeta = 100$ the interface does not move when $\\epsilon = 0.5$ and moves only slightly when $\\epsilon = 0.8$.}\n\t\\label{fig:flat-param}\n\\end{figure}\n\n\nFigure~\\ref{fig:flat-param-b} shows the evolution of the same initial configuration and $t=90$ but for $\\zeta = 100$.  In this case the smectic region and the density remain almost the same as in the initial condition shown in Fig.~\\ref{fig:flat-ic}. The reason why further growth is not observed can be explained by the effective bifurcation parameter $\\epsilon_e$ in Eq.~(\\ref{eq:psi-const2}). If the smectic grows and the smectic-isotropic interface advances into regions of lower density, we argue that $\\epsilon_e$ increases over $\\epsilon$ in such regions. Hence when $\\zeta$ is large enough, the effective bifurcation parameter can increase even beyond the coexistence value $\\epsilon_c$, which prevents any growth of the smectic.\n\nSimilarly, we show in Figs.~\\ref{fig:flat-param-c} and \\ref{fig:flat-param-d} the case $\\epsilon = 0.8$ (within the isotropic region of the phase diagram) for $\\zeta = 1$ and $100$, at $t = 90$.  Consistent with the above discussion, we see some shrinkage of the smectic while increasing the density of the isotropic phase (satisfying the balance of mass) when $\\zeta=1$.  However when $\\zeta=100$ there is significantly slower evolution of the smectic because of the strong coupling between the order parameter and the density. This can also be explained in terms of the effective bifurcation parameter $\\epsilon_e$, as in this case if the smectic tries to shrink it creates a region where $\\epsilon_e$ decreases from $\\epsilon$.\n\n\n\\section{Morphological evolution and flows in focal conic domains}\n\\label{sec:fcflow}\n\nThe outer boundary of a FCD in smectic films has a mean curvature $H$ that changes sign from positive away from the center, to negative near the center. At that point there is a macroscopic singularity in the form of a cusp. Near the cusp, the principal curvatures become very large, which can lead to interesting interfacial stresses and flows. Experiments have shown that the FCD undergoes complex morphological changes under temperatures changes, mediated by evaporation of smectic layers. In this section, we study the velocity field in a smectic-isotropic system presenting layers bent in a focal conic configuration. We also revisit the morphological transitions studied in~\\cite{vitral2019role}, which were previously investigated within a purely diffusional model (governed by the order parameter equation only), and for uniform density. First, we show that for a certain range of parameters, the transition from focal conic defects to conical pyramids or concentric rings is also observed in the weakly compressible model with density contrast. We then investigate how the velocity field changes during these morphological transitions, revealing the roles of Gaussian curvature $G$ and layer orientation at the interface on flow fields.\n\n\\subsection{Flows in focal conics at coexistence}\n\nWe analyze the velocity fields obtained at coexistence $\\epsilon_c = 0.675$ for different values of $\\zeta$ and density ratios, using an initial condition consisting of bent layers forming a focal conic (see Fig.~\\ref{fig:fc-cp} left panel). The computational domain is a cubic cell with $N = 256^3$ grid points, with approximately 8 points per wavelength and $q_0 = 1$, so that $L_x = 200$, $L_y = 200$ and $L_z = 200$. Parameters used are $\\beta = 2$, $\\gamma = 1$, and $\\nu = \\lambda = 1$. We will compare the velocity field $\\mathbf{v}$ obtained when $\\zeta$ is large and the system approaches quasi-incompressibility with that obtained when $ \\zeta$ is small.\n\nAs a reference, we begin with the case with $\\kappa = 0$, so $\\rho_s = \\rho_0 = 1$. Figure~\\ref{fig:fcflow1} shows $\\mathbf{v}$ at time $t = 4$ at the mid section of the domain, $y = L_y\/2$, for $\\zeta = 100$ and $\\zeta = 0.01$. As expected, for $\\zeta = 100$ the flow behaves similarly to an incompressible uniform density system~\\cite{vitral2020model}.  Vortices appear at the smectic-isotropic interface, where the flow moves outward from the smectic in regions of negative mean curvature, and inward towards the smectic in regions of positive mean curvature. The flow is governed by local curvatures since no significant density deviation from $\\rho = 1$ is observed due to the large value of $\\zeta$. The term $\\bar{\\mu}\\nabla\\psi$ determines $\\mathbf{v}$ in Eq.~(\\ref{eq:wcom-stokes2}), where the local curvatures appear from the difference in chemical potential $\\mu$ between a planar and a curved interface. This difference is given by the Gibbs-Thomson equation for the case of layers parallel to the interface~\\cite{vitral2019role}:\n\\begin{eqnarray}\n    \\delta\\mu\\Delta A &=& 2H\\sigma_h + (4H^2-2G)\\sigma_b - 2H(3G-4H^2)\\sigma_t \\;.\n    \\label{eq:gibbs-thomson}\n\\end{eqnarray}\nHere $\\Delta A$ is the difference in amplitude between the two phases, $\\sigma_h$ is the surface tension, $\\sigma_b$ the interface bending coefficient, and $\\sigma_t$ the interface torsion coefficient. These coefficients can be directly computed from the model parameters, see~\\cite{vitral2019role}.\n\nFor surfaces where the leading order term proportional to $H$ dominates Eq.~(\\ref{eq:gibbs-thomson}), the difference in chemical potential $\\delta\\mu$ is positive for $H > 0$, implying in a thermodynamic force $\\bar{\\mu}\\nabla\\psi$ pointing towards the smectic at the interface, and the opposite when $\\delta\\mu$ is negative. Based on Eq.~(\\ref{eq:wcom-stokes2}), this corroborates with the numerical results discussed for Fig.~\\ref{fig:fcflow1}(a), where we also notice that the flow is stronger in regions where the magnitude $H$ is high. However, when $\\zeta$ is reduced to $\\zeta = 0.01$, the flow in the isotropic phase at $t = 4$ simply points away from the interface, whereas inside the smectic the flow also moves up, and curves near the interface. The reason is that for small $\\zeta$ the density has more freedom to deviate from the equilibrium values $\\rho_0$ and $\\rho_s$, particularly at the interface.  This creates density gradients that interfere with the resulting flow structure.\n\n\\begin{figure}[ht]\n\t\\centering\n    \\begin{subfigure}[b]{0.46\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/fc-psiQuiver-rho1to1-zt100-e0d675-r140-nu1-t4.pdf}\n    \\caption{$\\zeta = 100$}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.46\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/fc-psiQuiver-rho1to1-zt0d01-e0d675-r140-nu1-t4.pdf}\n    \\caption{$\\zeta = 0.01$}\n    \\end{subfigure}\n    \\caption{Comparison between the transient fluid flow $\\mathbf{v}$ on smectic-isotropic fluid system for different $\\zeta$, at an early time $t = 4$, where both phases have the same bulk density ($\\kappa = 0$ and $\\rho_0 = 1$). Background color is the order parameter $\\psi$. We use $N = 256^3$, $\\Delta t = 1\\times 10^{-3}$, and parameters $q_0=1$, $\\eta=1$, $\\epsilon = 0.675$ (coexistence), $\\alpha=1$, $\\beta=2$ and $\\gamma=1$.}\n\t\\label{fig:fcflow1}\n\\end{figure}\n\nWe now increase the density contrast to $\\rho_s:\\rho_0 =  2:1$, using $\\kappa =  0.3727$ and $\\rho_0 = 0.5$. Figure~\\ref{fig:fcflow2} shows numerical results for $\\mathbf{v}$ at time $t = 4$ for the cases of $\\zeta = 100$ and $\\zeta = 0.01$. When compared to Fig.~\\ref{fig:fcflow1}(a), the velocity field for $\\zeta = 100$ presents smaller vortices closer to the smectic region, which quickly decay away from the interface in the isotropic phase. This is characteristic of quasi-incompressibility, since for large $\\zeta$ the density is constant almost everywhere except at the sharp interface. For the case of $\\zeta = 0.01$, the velocity points upwards throughout the smectic-isotropic system, implying that the density gradient dominates the orientation of the flow. This result is similar to Fig.~\\ref{fig:fcflow1}(b), although for Fig.~\\ref{fig:fcflow2}(b) the flow in the smectic is stronger; this happens owing to the combination of the larger density gradient for $\\rho_s:\\rho_0 =  2:1$ with the small value of $\\zeta$.  In this case the density at the interface becomes more diffuse, enhancing the longitudinal flow that moves from regions of high to low density.\n \n\\begin{figure}[ht]\n\t\\centering\n    \\begin{subfigure}[b]{0.46\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/fc-psiQuiver-rho2to1-zt100-e0d675-r140-nu1-t4.pdf}\n    \\caption{$\\zeta = 100$}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.46\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/fc-psiQuiver-rho2to1-zt0d01-e0d675-r140-nu1-t4.pdf}\n    \\caption{$\\zeta = 0.01$}\n    \\end{subfigure}\n    \\caption{Comparison between the transient fluid flow $\\mathbf{v}$ on smectic-isotropic fluid system for different $\\zeta$, at an early time $t = 4$, with $\\rho_s:\\rho_0 = 2:1$ density ratio between bulk phases ($\\kappa = 0.3727$ and $\\rho_0 = 0.5$). Background color is the order parameter $\\psi$. We use $N = 256^3$, $\\Delta t = 1\\times 10^{-3}$, and parameters $q_0=1$, $\\eta=1$, $\\epsilon = 0.675$ (coexistence), $\\alpha=1$, $\\beta=2$ and $\\gamma=1$.}\n\t\\label{fig:fcflow2}\n\\end{figure}\n\nOn increasing the density ratio to $\\rho_s:\\rho_0 = 100:1$, by using $\\kappa = 0.7379$ and $\\rho_0 = 0.01$, some significant changes in the velocity are observed for large $\\zeta$. Figure~\\ref{fig:fcflow3-a} shows that for $\\zeta = 100$ at $t = 4$, the flow in the smectic becomes dominated by the potential part of the velocity, so that it points radially outward from the layers in the direction of the density gradient. The velocity in the isotropic phase is smaller in magnitude, and points towards the smectic. This happens because for large $\\zeta$ the density gradient becomes large at the interface owing to the large density ratio. We also show in Fig.~\\ref{fig:fcflow3-c} the transient flow for $\\zeta = 100$ at a later time, $t = 25$.  The flow in the smectic has mostly disappeared, so the only significant flow is in the isotropic phase towards the smectic and tangential at the interface. Since the density of the isotropic phase is small, the mass flux in this case becomes very small in magnitude and decreases with time.\n\nFor $\\zeta = 0.01$ Fig.~\\ref{fig:fcflow3-b} shows that the velocity at $t = 4$ points upwards as in Fig.~\\ref{fig:fcflow2}(b). In Fig.~\\ref{fig:fcflow3-d}, we see that the velocity at $t = 25$ is still primarily pointing upwards, and that significant growth of smectic layers at the interface has taken place when since $t = 4$.  Since the energy penalty for deviations of the preferred bulk density is small for a small $\\zeta$, the mass flux $\\rho\\mathbf{v}$ from the bulk smectic towards the interface leads to the growth of layers. That is, growth of the smectic by mass flow may occur, albeit at the cost of reducing the bulk smectic density in order to satisfy overall mass conservation.\n\n\\begin{figure}[ht!]\n\t\\centering\n    \\begin{subfigure}[b]{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/fc-psiQuiver-rho100to1-zt100-e0d675-r140-nu1-t4.pdf}\n    \\caption{$t = 4$, $\\zeta = 100$}\n    \\label{fig:fcflow3-a}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/fc-psiQuiver-rho100to1-zt0d01-e0d675-r140-nu1-t4.pdf}\n    \\caption{$t = 4$, $\\zeta = 0.01$}\n    \\label{fig:fcflow3-b}\n\\end{subfigure}\n    \\begin{subfigure}[b]{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/fc-psiQuiver-rho100to1-zt100-e0d675-r140-nu1-t25.pdf}\n    \\caption{$t = 25$, $\\zeta = 100$}\n    \\label{fig:fcflow3-c}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/fc-psiQuiver-rho100to1-zt0d01-e0d675-r140-nu1-t25.pdf}\n    \\caption{$t = 25$, $\\zeta = 0.01$}\n    \\label{fig:fcflow3-d}\n    \\end{subfigure}\n    \\caption{Comparison between the transient fluid flow $\\mathbf{v}$ on smectic-isotropic fluid system for different $\\zeta$, at times $t = 4$ and $t = 25$, with $\\rho_s:\\rho_0 = 100:1$ density ratio between bulk phases ($\\kappa = 0.7379$ and $\\rho_0 = 0.01$). Background color is the order parameter $\\psi$. We use $N = 256^3$, $\\Delta t = 1\\times 10^{-3}$, and parameters $q_0=1$, $\\eta=1$, $\\epsilon = 0.675$ (coexistence), $\\alpha=1$, $\\beta=2$ and $\\gamma=1$.}\n\t\\label{fig:fcflow3}\n\\end{figure}\n\n\n\\subsection{Flows in focal conics under thermal sintering}\n\nTransitions in smectic thin films from focal conic domains to conical pyramids through sintering have been experimentally observed by Kim et al.~\\cite{kim2016controlling,kim2018curvatures}. By increasing the value of $\\epsilon$ (favoring the isotropic phase) we can simulate a heat treatment of a smectic film similar to the sintering experiments. We set as initial condition the configuration of Fig.~\\ref{fig:fc-cp} (left), with smectic layers bent in a focal conic configuration, in a domain with $N = 256^3$ grid points, so that $L_x = 200$, $L_y = 200$ and $L_z = 200$. Parameter values used are $\\beta = 2$, $\\gamma = 1$, $\\nu = \\lambda = 100$, $\\rho_0 = 0.5$, $\\rho_s = 1$ and $\\zeta = 0.01$. We set $\\epsilon = 0.8$, so that the initial smectic region should slowly evaporate at the interface with the isotropic phase. We choose a small value for $\\zeta$ in order to allow the transition to take place. If $\\zeta$ is too large the focal conic morphology is unable to change significantly since the motion of $\\psi$ becomes restricted due to the balance of mass. Figure~\\ref{fig:fc-cp} shows the initial focal conic (left) and the resulting conical pyramid (right) at time $t = 80$, after a number of layers have evaporated away from the core, sculpting a pyramidal structure that is more resilient to evaporation than the original focal conic. Also, the velocity field for a focal conic under $\\epsilon = 0.8$ initially points away from the smectic, similarly to Fig.~\\ref{fig:fcflow2}(b).\n\nNote that the edges of the smectic layers in the pyramidal region become exposed at the interface. As discussed in~\\cite{vitral2019role}, the equation describing interfacial thermodynamics for this perpendicular alignment of layers with respect to the interface is different from the classical Gibbs-Thomson equation found in literature, even at leading order. The difference in chemical potential between planar and curved interfaces in this case is given by\n\\begin{eqnarray}\n  \\delta\\mu \\Delta A &=& \\bigg[\\frac{1}{2}\\nabla^2_s H + 2H(H^2 - G)\\bigg] \\frac{\\sigma_h}{q_0^2} \\;,\n  \\label{eq:gt-perp}\n\\end{eqnarray}\nwhere the combination of curvatures in the RHS is similar to Willmore-type flows~\\cite{willmore1996riemannian}. Therefore, we consider the effect of layer alignment at the interface on flow.\n\nFor a small dimensionless $\\zeta$, local curvatures dominate the motion of $\\psi$. A small $\\zeta$ also leads to a very diffuse density variation at the interface, which evolves slowly compared to the order parameter. Based on the balance of linear momentum in Eq.~(\\ref{eq:blm-ge-w}), under these conditions the velocity at the interface of a conical pyramid is governed by the force $\\bar{\\mu}\\nabla\\psi$. We have previously shown~\\cite{vitral2019role} that conical pyramids are surfaces for which the RHS of Eq.~(\\ref{eq:gt-perp}) is approximately zero, and present values of $H^2$ and $G$ that are locally close to each other along most of the surface. That is, conical pyramids in this context are analogous to Willmore surfaces~\\cite{willmore1996riemannian}, for which the Willmore flow is zero. This has an important implication to the normal force balance at the interface, as we now show.\n\n\\begin{figure}[ht]\n\t\\centering\n    \\begin{subfigure}[b]{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/focalconic.png}\n    \\end{subfigure}\n    \\hspace{10mm}\n    \\begin{subfigure}[b]{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/psi-t50-cpy512-2.png}\n    \\end{subfigure}\n    \\caption{Focal conic (initial condition) transforms into a conical pyramid ($t = 80$) through evaporation of layers. Red and blue planes represent $+1$ and $-1$ values of $\\psi$, respectively. Parameters used are $\\epsilon = 0.8$ (favoring the isotropic phase), $\\beta = 2$, $\\gamma = 1$, $\\nu = 100$, $\\lambda = 100$, $\\rho_0 = 0.5$, $\\kappa = 0.3727$ ($\\rho_s = 1$), and $\\zeta = 0.01$.}\n\t\\label{fig:fc-cp}\n\\end{figure}\n\nWhen layers are perpendicular to the interface, $\\nabla\\psi$ has both a tangential and a normal component at this interface, where the normal component is the same as the gradient of the amplitude $\\nabla A$ at this interface (which only acts on the slow spatial scale~\\cite{vitral2020spiral}). Since Eq.~(\\ref{eq:gt-perp}) is derived from the amplitude equation that governs the motion of $A$, we find that $\\delta\\mu \\Delta A$ corresponds to the asymptotic form of the force $\\bar{\\mu}\\nabla\\psi$ in the normal direction to the curved interface. \n\nThe fact that the RHS of Eq.~(\\ref{eq:gt-perp}) becomes very small for a conical pyramid then suggests that the normal velocity $\\mathbf{v}\\cdot\\mathbf{n}$ at the interface is negligible when compared to tangential flow. In Fig.~\\ref{fig:cp-flow}, we show the middle cross section of the $\\psi$ field for a conical pyramid alongside the velocity field. As expected, the numerical result reveals that the flow indeed becomes tangential on the surface of the pyramid, reinforcing that mean and Gaussian curvatures balance in such a way that the RHS of Eq.~(\\ref{eq:gt-perp}) and normal forces become negligible on this surface.\n\n\\begin{figure}[ht]\n\t\\centering\n    \\includegraphics[width=0.45\\textwidth,height=0.42\\textwidth]{figures\/cpyr-quiver-rho2to1-e0d8-nu100-z0d01-t80.pdf}\n    \\caption{Order parameter field for a conical pyramid (in cross-section) at $t = 80$ obtained from an initial focal conic configuration. The velocity field $\\mathbf{v}$ is also plotted, showing that the velocity becomes tangential to the interface of the pyramid. Parameters are $\\epsilon = 0.8$ (favoring the isotropic phase), $\\beta = 2$, $\\gamma = 1$, $\\nu = 100$, $\\lambda = 100$, $\\rho_0 = 0.5$, $\\kappa = 0.3727$, and $\\zeta = 0.01$.}\n\t\\label{fig:cp-flow}\n\\end{figure}\n\n\n\\section{Domain interactions in smectic-isotropic systems}\n\\label{sec:domain}\n\nWe consider two applications of the weakly compressible smectic-isotropic model that involve interactions between two domains. The first involves examining the coalescence of neighboring cylindrical stacks of smectic layers, which is motivated by experiments in freely-suspended smectic films. The second is an investigation of the interactions between two FCDs, once again motivated by the experimental observations from Kim et al.~\\cite{kim2016controlling,kim2018curvatures}.\n\n\\subsection{Coalescence of smectic cylindrical domains}\n\nDomain coalescence driven by capillarity is a widely studied phenomenon in fluid dynamics. For two infinite isotropic fluid cylinders with the same radius, Hopper~\\cite{hopper1984coalescence,hopper1993coalescence} developed an exact theory based on the assumption of creeping planar viscous flow, parameterizing the coalescence in the plane through a 1-parameter family of closed inverse ellipses of constant area. This theory has been shown to be qualitatively consistent with a bridge growing between two smectic islands~\\cite{shuravin2019coalescence,nguyen2020coalescence}, although with a slower temporal evolution. In a thin smectic-A liquid crystal equilibrium, which is freely suspended, islands are regions with more smectic layers than the embedding film, resulting in smectic cylinders enveloped by outer layers, bounded by edge dislocation loops. The line tension arising from the dislocations is argued to drive the initial bridge expansion in the coalescence process. In contrast to ordinary fluids, Nguyen et al.~\\cite{nguyen2020coalescence,nguyen2011smectic} showed that permeation through the molecular layers of the merging islands is an additional channel for dissipative motion, not included in Hopper's model, and it may be responsible for the observed slow coarsening dynamics. Coalescence of smectic islands~\\cite{shuravin2019coalescence,nguyen2020coalescence} and holes~\\cite{dolganov2020coalescence} have been observed in freely-suspended smectic films (FSSF) with layers parallel to the surface of the film, with thicknesses that can range from two layers up to thousands of layers.\n\nHere, we study numerically a related configuration consisting of two smectic cylinders, which are initially touching, surrounded by an isotropic phase of different density. While this geometry is much simpler than a FSSF, it connects to Hopper's work on the coalescence of two fluid cylinders, and also shows the role played by irrotational flow in our model. As smectic is a uniaxial phase, we expect to observe qualitatively different features in the interaction between two smectic cylinders when compared to isotropic fluids. We consider a $N = 256^2\\times32$ grid as the computational domain, so that $L_x = 200$, $L_y = 200$ and $L_z = 25$. The domain contains two smectic cylinders, each with radius $R_0 = 37.5$ and five layers, embedded in the middle of an isotropic domain. Parameters used are $\\beta = 2$, $\\gamma = 1$, $\\nu = \\lambda = 1$, $\\rho_0 = 0.5$, $\\kappa = 0.3727$ (2:1 density ratio) and $\\zeta = 100$. We choose $\\epsilon = 0.3$, deep in the smectic state (as $\\epsilon_c = 0.675$), so that the smectic is energetically favored. At the same time, we take $\\zeta = 100$ to be high enough to guarantee a weak conservation of $\\psi$, a necessary condition for smectic coalescence to take place (if $\\zeta$ is too small, $\\psi$ would grow and occupy the whole domain with $\\epsilon = 0.3$ and a 2:1 density ratio). Figure~\\ref{fig:cls3d} shows the two smectic cylinders creating a bridge between each other at time $t = 2.5$, and also the resulting stationary cylinder at time $t = 200$ after coalescence occurs. Note that in addition to a two-dimensional coalescence process in the plane of the smectic layers, we observe a number of layers growing in the normal direction $\\hat{z}$, from five initial layers to eight in the resulting cylinder (filling the whole $L_z$). \n\n\\begin{figure}[ht]\n\t\\centering\n    \\begin{subfigure}[b]{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/pancake2.pdf}\n    \\caption{$t = 2.5$}\n    \\end{subfigure}\n    \\hspace{10mm}\n    \\begin{subfigure}[b]{0.4\\textwidth}\n    \\includegraphics[width=\\textwidth]{figures\/pancake-final2.pdf}\n    \\caption{$t = 200$}\n    \\end{subfigure}\n    \\caption{Coalescence of smectic cylinders inside a box with $L_x = L_y = 200$ and $L_z  = 25$. Besides coalescence, the number of layers of the final cylinder grow with respect to the initial ones. Parameters are $\\epsilon = 0.3$ (smectic region), $\\zeta = 100$, $\\beta = 2$, $\\gamma = 1$, $\\nu = 1$, $\\lambda = 1$, $\\rho_s = 1$, and $\\rho_0 = 0.5$ ($\\kappa = 0.3727$).}\n\t\\label{fig:cls3d}\n\\end{figure}\n\n\nIn order to make contact with Hopper's theory for infinite fluid cylinders, we focus on the two-dimensional dynamics on the plane of the stacks by changing the initial condition from five to eight smectic layers, so that both cylinders occupy the entire $z$ range of the domain. This initial condition is intended to mimic Hopper's case of infinite cylinders (and is also closer to the smectic islands constrained by enveloping layers observed in experiments). Figure~\\ref{fig:cls2d} shows the evolution of order parameter on the midplane ($L_z\/2$) of the box: due to the large $\\zeta$, the area defined by $\\psi$ shows little growth as it evolves. That is, as $\\zeta$ and the density ratio gets larger, the area becomes more closely conserved in its evolution. The midplane density evolves in the same way, since the motion of $\\psi$ is tied to $\\rho$ for large $\\zeta$.\n\n\\begin{figure}[ht]\n\t\\centering\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/psiXY-h32-rho2to1-e0d2-nu1-t0d5.pdf}\n        \\caption{$t = 0.5$}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/psiXY-h32-rho2to1-e0d2-nu1-t10.pdf}\n        \\caption{$t = 10$}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/psiXY-h32-rho2to1-e0d2-nu1-t50.pdf}\n        \\caption{$t = 25$}        \n    \\end{subfigure}\n    \\caption{Order parameter at the mid height $xy$ plane of a box with $L_x = L_y = 200$ and $L_z  = 25$ showing coalescence of smectic cylinders, at times $t = 0.5$, $t = 10$ and $t = 50$. Initial conditions present two parallel smectic cylinders with layers filling the whole height $L_z$. Parameters are $\\epsilon = 0.3$ (smectic region), $\\zeta = 100$, $\\beta = 2$, $\\gamma = 1$, $\\nu = 1$, $\\lambda = 1$, $\\rho_s = 1$, and $\\rho_0 = 0.5$ ($\\kappa = 0.3727$).}\n    \\label{fig:cls2d}    \n\\end{figure}\n\nWhile capillarity driven coalescence is well understood for objects such as droplets or cylinders of isotropic fluids, the modulated nature of a smectic leads to several significant differences.  First, the interface between the cylinder and the isotropic phase is composed of layers oriented perpendicular to the interface. According to the equations derived in~\\cite{vitral2019role}, shown here in Eq.~(\\ref{eq:gt-perp}), diffusion driven interface motion of the smectic for such orientation goes as $v_n \\sim -H^{3}$ for zero Gaussian curvature (as in a cylinder), which by itself leads to coalescence at a slower rate than the classical motion by mean curvature. The smectic motion also depends on the evolution of the density, $\\dot{\\rho}$, governed by Eq.~(\\ref{eq:bm-ge-w}), which is proportional to $-\\nabla\\cdot\\mathbf{v}$. Therefore, in the coalescence of smectic cylinders, a competition exists between the diffusional motion of the order parameter and mass transport.\n\nRecall that the divergence of the velocity depends on the irrotational flow, $\\nabla\\cdot\\mathbf{v} = \\nabla^2\\Phi$, given by Eq.~(\\ref{eq:irrot}) in Fourier space. The first term on the RHS of Eq.~(\\ref{eq:irrot}) is proportional to the normal $\\mathbf{n}$ at the interface when $\\rho \\neq \\rho_0 + \\kappa A$. Since $\\nabla\\cdot\\mathbf{n} = 2H$, this implies that $\\dot{\\rho}$ is proportional to the negative of the mean curvature, so that there is a mean curvature driven flow of mass from regions of positive to negative curvature. To verify this, we plot the divergence of the velocity for time $t = 5$ in Fig.~\\ref{fig:divv}. Regions of positive $\\nabla\\cdot\\mathbf{v}$ correspond to an interface of positive mean curvature, while regions of negative $\\nabla\\cdot\\mathbf{v}$ have an interface of negative mean curvature. Figure~\\ref{fig:divv} shows the enlarged bridge region, and the velocity field $\\mathbf{v}$. During coalescence, the flow in the left cylinder moves toward the right, and vice-versa, and at the bridge the flow moves inwards in the region of negative mean curvature, as expected from the coalescence process. \n\n\\begin{figure}[ht]\n\t\\centering\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/divv-rho2to1-e0d3-z100-nu1-t5.pdf}\n    \\end{subfigure}\n    \\hspace{10mm}\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/divvq-zoom-rho2to1-e0d3-z100-nu1-t5.pdf}\n    \\end{subfigure}\n    \\caption{Divergence of the velocity $\\nabla\\cdot\\mathbf{v}$ at the mid height $xy$ plane of a box with $L_x = L_y = 200$ and $L_z  = 25$ at time $t = 5$. Initial conditions and parameters are the same as in Fig. \\ref{fig:cls2d}. The right figure is a close up of the bridge region, showing that the velocity field $\\mathbf{v}$ moves from regions of positive $\\nabla\\cdot\\mathbf{v}$ ($H > 0$) towards regions\n    of negative $\\nabla\\cdot\\mathbf{v}$ ($H < 0$).}\n    \\label{fig:divv}    \n\\end{figure}\n\nIn our numerical calculations involving smectic cylinders spanning the domain in $z$, we observe coalescence deep inside the smectic region from $\\epsilon = 0.2$ up to the coexistence point \\mbox{$\\epsilon_c = 0.675$}. In Fig.~\\ref{fig:bridge} we plot the normalized bridge length between the smectic stacks as a function of time for different values of $\\epsilon$. Coalescence occurs faster as we decrease the bifurcation parameter from the coexistence value, that is, as we decrease the energy of the smectic in comparison to the isotropic phase (corresponding to a decrease in temperature). Qualitatively these curves and the numerical evolution of the order parameter resemble the bridge width evolution shown in experiments and Hopper's theoretical model. The phase-field model also allows for more intricate order parameter morphologies, so that a possibility for future work is to simulate FSSF and study the role of permeation on coalescence.\n\n\\begin{figure}[ht]\n\t\\centering\n    \\includegraphics[width=0.44\\textwidth]{figures\/bridge-width-rho2to1-z100-nu1.png}\n    \\caption{Bridge width as a function of time, for simulations employing different values of $\\epsilon$. Domain size, initial conditions and parameters are the same as in Fig. \\ref{fig:cls2d}.}\n\t\\label{fig:bridge}\n\\end{figure}\n\nInterestingly, if the initial smectic cylinders do not span the computational domain in the $z$-direction, coalescence is only observed once $\\epsilon$ becomes sufficiently small. For example, using a domain with $N = 256^2\\times64$, and the same parameters as above, smectic domains of approximately five layers did not coalesce for $\\epsilon$ close to $\\epsilon_c = 0.675$. From plots of the midplane order parameter in Fig.~\\ref{fig:sep}, we observe that after an initial thin bridge is formed (at $t = 25$), the two stacks move apart, until the bridge breaks (at $t = 35$). In this case, since five layers don't span the domain in the $z$-direction, growth in that direction competes with coalescence, as the total mass of the smectic phase is approximately conserved for large $\\zeta$. In order to help visualize this process, Fig.~\\ref{fig:sep-flow} shows both the velocity $\\mathbf{v}$ and the order parameter in the $x-z$ plane at $L_y\/2$. At early times, $t = 2.5$, the two cylinders are still closely in contact (at $x = 100$), but a flow forms that induces mass to move from the lateral of the cylinder towards the top and bottom. At a later time, $t = 25$, the flow continues to point opposite to the bridge (which is very thin at this point, as seen in Fig.~\\ref{fig:sep}), leading the cylinders to further separate from each other. As we have seen in Fig.~\\ref{fig:bridge}, coalescence is slower at a higher $\\epsilon$, so at $\\epsilon_c$ the coalescence bridge collapses as mass continuously moves away from it. At the top and bottom of the cylinders, we observe the formation of target structures~\\cite{knobloch2015spatial}, which are interfacial rings  (the bottom and top droplets shown in the cross-section) induced by the circular geometry of the order parameter. Therefore, while we argue that the capillary induced diffusion leads to a slow coalescence process, in this case the flow dominates the motion of the smectic and inhibits coalescence.\n\n\\begin{figure}[ht]\n\t\\centering\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/psiXY-h64-rho2to1-e0d675-nu1-t25.pdf}\n        \\caption{$t = 25$}        \n    \\end{subfigure}\n    \\hspace{10mm}\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/psiXY-h64-rho2to1-e0d675-nu1-t35.pdf}\n        \\caption{$t = 35$}\n    \\end{subfigure}\n    \\caption{Order parameter on the mid $xy$ plane of a domain with $L_x = L_y = 200$ and $L_z  = 50$ showing separation of smectic cylinders, at times $t = 2.5$ and $t = 25$. Initial conditions are two smectic tangential cylinders with approximately five layers. Parameters are $\\epsilon = 0.675$ (coexistence), $\\zeta = 100$, $\\beta = 2$, $\\gamma = 1$, $\\nu = 1$, $\\lambda = 1$, $\\rho_s = 1$, and $\\rho_0 = 0.5$ ($\\kappa = 0.3727$).}\n    \\label{fig:sep}    \n\\end{figure}\n\n\\begin{figure}[ht]\n\t\\centering\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/psiXZ-quiver-h64-rho2to1-e0d675-nu1-t2d5.pdf}\n        \\caption{$t = 2.5$}\n    \\end{subfigure}\n    \\hspace{10mm}\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/psiXZ-quiver-h64-rho2to1-e0d675-nu1-t25.pdf}\n        \\caption{$t = 25$}\n    \\end{subfigure}\n    \\caption{Order parameter and velocity field on the mid $xz$ plane for the same calculation of Fig.~\\ref{fig:sep}. The $z$ axis has been multipled by a factor of four for better visualizing the flow. Left: two smectic cylinder in contact at $x = 100$, and time $t = 2.5$. Right: smectic cylinders are almost separating due to the flow, time $t = 25$.}\n    \\label{fig:sep-flow}    \n\\end{figure}\n\nWe note finally that coalescence does not occur in the isotropic region $\\epsilon > \\epsilon_c$ for any of the geometries shown in this section, even for cylindrical domains spanning the computational cell. Numerical results are shown in Fig.~\\ref{fig:e0d9}, using the same parameters and initial configuration as in Fig.~\\ref{fig:cls2d}, with $\\epsilon = 0.9$. The smectic region shrinks since $\\epsilon > \\epsilon_c$. No bridge is initially formed, and the cross-section of the cylinders starts to melt into a target shape, as seen at $t = 7.5$. At time $t = 20$ the target cross-section further breaks into droplets, and at time $t = 200$ the order parameter field shows droplets spread all over the domain. The melting of smectic order when heating suspended films and formation of isotropic or nematic droplets have also been observed experimentally~\\cite{schuring2002isotropic,klopp2019structure}, so that the present model may provide further insights on the interaction, arrangement and dynamics of these droplets. In terms of differential equations, these results are strikingly different from classical Swift-Hohenberg dynamics (pure diffusional dynamics of $\\psi$), for which the order parameter in the isotropic region would simply disappear in time. We conclude that coalescence requires synergy between flow and diffusive dynamics of the order parameter and density, and it may not occur if one of them becomes antagonistic to the coalescence process.\n\n\\begin{figure}[ht]\n\t\\centering\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/psiXY-h32-rho2to1-e0d9-nu1-t7d5.pdf}\n        \\caption{$t = 7.5$}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/psiXY-h32-rho2to1-e0d9-nu1-t20.pdf}\n        \\caption{$t = 20$}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/psiXY-h32-rho2to1-e0d9-nu1-t200.pdf}\n        \\caption{$t = 200$}\n    \\end{subfigure}\n    \\caption{Order parameter at the mid height $xy$ plane of a box with $L_x = L_y = 200$ and $L_z  = 25$ showing disintegration of smectic cylinders, at times $t = 7.5$, $t = 20$ and $t = 200$ (steady state). Initial conditions present two smectic tangential cylinders with layers filling the height $L_z$. Parameters are $\\epsilon = 0.9$ (isotropic region), $\\zeta = 100$, $\\beta = 2$, $\\gamma = 1$, $\\nu = 1$, $\\lambda = 1$, $\\rho_s = 1$, and $\\rho_0 = 0.5$ ($\\kappa = 0.3727$).}\n    \\label{fig:e0d9}    \n\\end{figure}\n\n\n\n\\subsection{Interactions between focal conic domains}\n\nWhile topological defects are known to interact in various soft matter systems, little is known about interactions between focal conics, or even about flows on their surface. The formation of a focal conic domain of the type investigated in this paper depends on the balance between the splay energy to form the conic (proportional to splay elastic constant $K_{1}$), and the difference between the surface tensions for parallel and perpendicular molecular anchoring at the the smectic-air interface ($\\Delta \\sigma = \\sigma_{\\parallel}-\\sigma_{\\perp}$). The size of the conic is determined by the balance of the two energies, and it is of the order of $K_{1}\/|\\Delta \\sigma|$~\\cite{lavrentovich1994nucleation}. Once an equilibrium array of focal conics is formed, Kim et al.~\\cite{kim2016controlling} have reported experiments on sintering of FCDs that showed morphologies in which neighboring focal conic domains interact, for example via thin tunnel like structures that remains from the original film. In order to begin to understand these interactions, we have investigated numerically the evolution of initial configurations that consist of two focal conics, where we have biased the system away from equilibrium. We show three different examples here: (1) we differentially compress the smectic layers of two neighboring focal conic domains, (2) we impose a density gradient in the isotropic phase from one defect towards the other, and (3) we start from an initial condition in which two focal conics overlap. In all cases we use a computational domain with $N = 512 \\times 256^2$ grid points,  8 points per $\\psi$ wavelength and $q_0 = 1$, so that $L_x = 400$, $L_y = 200$ and $L_z = 200$, and we set the density ratio between phases $\\rho_s:\\rho_0$ as $2:1$ (with $\\rho_s = 1$ and $\\rho_0 = 0.5$). Other parameters used are $\\beta = 2$, $\\gamma = 1$, $\\nu = \\lambda = 1$, and $\\zeta = 1$, which allows for a moderate coupling between the order parameter and density fields.\n\n\\subsubsection{Layer compression}\n\nThe equilibrium layer spacing, given by $\\lambda_0\/2$, may be changed by straining the liquid crystal~\\cite{clark1973strain}. For arrays of FCDs, due to boundary conditions or proximity to defects, some regions may have layers deviating from $\\lambda_0\/2$. We impose a variable strain so that the local layer wavenumber increases linearly from $1.2q_0$ at $x = 0$ to $0.8q_0$ at $x = L_x$. Hence, the layers in the left focal conic are initially compressed, and the right ones stretched. Figure~\\ref{fig:2fc-stress} shows the velocity $\\mathbf{v}$ and order parameter $\\psi$ at the middle cross section of the domain ($y = L_y\/2$) for $\\epsilon = \\epsilon_c$. At early times, $t = 0.5$, there is induced flow from the conic in compression to the conic in extension. At $t = 15$ there is a strong flux in the bulk smectic so that the compressed conic grows toward the isotropic phase while increasing the interlayer spacing. On the other side, the conic with stretched layers contracts by decreasing the interlayer spacing. Simultaneous evaporation\/condensation and mass transport through the smectic allows the initially compressed (expanded) focal conic to expand (contract) at a later time, so that its layers achieve the required equilibrium layer spacing.\n\n\\begin{figure}[ht]\n\t\\centering\n    \\begin{subfigure}[b]{0.72\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/2fc-psiQuiver-rho2to1-stress-e0d675-r140-nu1-z1-t0d5.pdf}\n        \\caption{$t = 0.5$}\n\\end{subfigure}\n    \\begin{subfigure}[b]{0.72\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/2fc-psiQuiver-rho2to1-stress-e0d675-r140-nu1-z1-t15.pdf}\n        \\caption{$t = 15$}\n    \\end{subfigure}\n    \\caption{Two neighboring focal conics with a varying strain field, showing both the transient velocity field $\\mathbf{v}$ and the order parameter $\\psi$. The layers of the left focal conic are initially compressed, and those of the right focal conic stretched.}\n    \\label{fig:2fc-stress}    \n\\end{figure}\n\n\\subsubsection{Imposed density gradient in the isotropic phase}\n\nThe second situation we consider is an initial condition consisting of two identical focal conics, with an imposed density gradient in the isotropic phase. The initial density in the isotropic phase is a function of $x$, starting at $\\rho = 0.7$ at $x = 0$, and decreasing linearly up to $\\rho = 0.2$ at $x = L_x$ (hence deviating from $\\rho_0 = 0.5$).  The density of the smectic is constant at $\\rho_s = 1$. Figures~\\ref{fig:2fc-rhog} (a) and (b) show the velocity field with the density in the background for times $t = 0.5$ and $t = 25$ with $\\epsilon = 0.8$. The velocity field goes to the right (so from high density to low density) in the isotropic phase, and also points to the right in the smectic, despite the smectic having uniform density $\\rho_s = 1$.  At time $t = 25$, the velocity field is small in the smectic, but flow in the isotropic region continues since the density is still not homogeneous. Figure~\\ref{fig:2fc-rhog}(c) shows the order parameter field $\\psi$ at time $t = 25$. Note that the radius of the region containing the left focal conic has increased in time due to the imposed density gradient. The center of the left focal conic has also moved from $x = 100$ to the right, and this region now spans up to approximately $x=215$, thereby compressing the right focal conic.\n\n\\begin{figure}[htb!]\n\t\\centering\n    \\begin{subfigure}[b]{0.72\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/2fc-rhoQuiver-rho2to1-xrho0d7to0d2-e0d8-nu1-z1-t0d5.pdf}\n        \\caption{$\\rho$, $\\;t = 0.5$}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.72\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/2fc-rhoQuiver-rho2to1-xrho0d7to0d2-e0d8-nu1-z1-t25.pdf}\n        \\caption{$\\rho$, $\\;t = 25$}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.72\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/2fc-psiQuiver-rho2to1-xrho0d7to0d2-e0d8-nu1-z1-t25.pdf}\n        \\caption{$\\psi$, $\\;t = 25$}\n    \\end{subfigure}\n    \\caption{Two neighboring focal conics in contact with an isotropic phase, with an initial density gradient present in the isotropic phase. Figures show the transient velocity field $\\mathbf{v}$ alongside the density (a,b), and with the order parameter (c).}\n    \\label{fig:2fc-rhog}    \n\\end{figure}\n\n\\subsubsection{Overlapping focal conics}\n\nThe third example is that of two focal conics with a reduced distance between their centers. Figure \\ref{fig:2fc-over} shows the velocity and order parameter fields at the cross section $y = L_y\/2$ with $\\epsilon = \\epsilon_c$. The initial distance between the cores is $L = 150$, whereas the minimum distance under these conditions to isolate two focal conics is $L_e = 200$. This compression creates a region of high positive mean curvature between the two focal conics, inducing strong flow from the isotropic phase towards the smectic in that region. This flow persists for long times as the focal conic domains relax to equilibrium.\n\n\\begin{figure}[htb]\n\t\\centering\n    \\begin{subfigure}[b]{0.72\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/2fc-psiQuiver-rho2to1-midt-e0d675-r140-nu1-z1-t5.pdf}\n        \\caption{$t = 5$}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.72\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/2fc-psiQuiver-rho2to1-midt-e0d675-r140-nu1-z1-t38.pdf}\n        \\caption{$t = 38$}\n    \\end{subfigure}\n    \\caption{Two neighboring focal conics that overlap at $x = 200$, showing both the transient velocity field $\\mathbf{v}$ and the order parameter $\\psi$. The minimum non-overlapping distance between their cores is $L_e = 200$, while the distance employed as initial condition is $L = 150$.}\n    \\label{fig:2fc-over}    \n\\end{figure}\n\n\\section{Conclusions}\n\nA coupled phase-field and hydrodynamics model has been introduced to describe a weakly compressible two phase system consisting of a smectic (soft modulated phase) in contact with an isotropic fluid of different density (e.g. water, air or the own liquid crystal isotropic state). A non-conserved smectic order parameter is coupled to a conserved mass density so as to accommodate non-solenoidal flows near the smectic-isotropic boundary arising from a density contrast between the two phases. The model energy is a functional of the order parameter and its derivatives, and also includes coupling to a conserved density that has different values in bulk smectic and isotropic regions. For large values of the coupling coefficient, the order parameter becomes approximately conserved. This is the quasi-incompressible limit in which the density is constitutively related to the order parameter. However for smaller values of the coupling coefficient the variations of density and order parameter become independent. In real physical systems, this suggests a temperature dependence of the coupling coefficient: for instance, it would become lower at elevated temperatures in order to allow the smectic to melt\/evaporate independently of the density. The model fully incorporates surface driven flows due to local stresses that depend on boundary curvatures, and non-solenoidal flows that arise from density gradients.\n\n\nThe weakly compressible model has been used to describe morphological instabilities in smectic thin films, away from an equilibrium configuration comprising an array of focal conic domains. Experiments show that, upon sintering, focal conics decay into conical pyramids and concentric rings. For a single focal conic domain, our model leads to a transition between focal conics and conical pyramids upon increasing the temperature, mediated, as in the experiments, by localized evaporation of smectic layers. Flows are induced by boundary stresses due to curvature -including Gaussian curvature- through an extended form of equilibrium thermodynamics conditions at a curved surface (the Gibbs-Thomson equation). Furthermore, different boundary conditions apply when smectic layers become exposed in pyramidal domains, which do not have a counterpart in classic thermodynamics. This configuration is associated with tangential flows at the boundary of conical pyramids, which helps explain why this structures persist while focal conics evaporate. Irrotational flows are also induced by boundary motion due to phase density changes. \n\nAs further applications, we consider configurations that mimic focal conic interactions. These include setting up a density gradient in the isotropic phase above two focal conics, adding a nonuniform strain in the smectic layers, and taking the inter-center distance of two conics small enough so the conics overlap. Finally, we discuss the coalescence of cylindrical stacks of smectic layers, motivated by Hopper's theory of coarsening of isotropic fluid cylinders, and experiments on freely-suspended smectic films. Coalescence depends on the competition between capillarity induced diffusion of the order parameter, local curvatures, and mass transport. Instead, if we simulate a heating of the smectic cylinders, numerical results show the melting of smectic order and formation of droplets, a phenomenon that has also been observed in experiments. Future work will address the complex role of permeation in the coalescence of islands and holes, as discussed in recent experiments, and further explore the physics behind the transition into arranged droplets.\n\n\n\\section*{Acknowledgments}\n\nThis research has been supported by the Minnesota Supercomputing Institute, and by the Extreme Science and Engineering Discovery Environment (XSEDE) \\cite{towns2014xsede}, which is supported by the National Science Foundation under Grant No. ACI-1548562. EV thanks the support from the Doctoral Dissertation Fellowship and from the Aerospace Engineering and Mechanics department, University of Minnesota. This research is also supported by the National Science Foundation under Grant No. DMR-1838977. Part of the work of JV was done during the ACTIVE20 program at the Kavli Institute for Theoretical Physics which is funded by the National Science Foundation under Grant No. NSF PHY-1748958.\n\n\n\\bibliographystyle{vancouver}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA research program on Arens-Michael envelopes and homological\nepimorphisms was initiated by Joseph Taylor in his seminal papers\n\\cite{T1,T2}.  Taylor was inspired by his previous results on\nmulti-operator holomorphic functional  calculus and some\nconsideration that can be incorporated in those circle of ideas\nthat is called Noncommutative Geometry nowadays. Homological\nepimorphisms play important role in modern attempts to find\ngeneralizations of Taylor spectrum and functional calculus for\nnoncommutative algebras (see \\cite{Do10A,Do10B}).\n\nWe pursue two aims: to get an explicit description of the\nArens-Michael envelope for $U(\\fg)$ and prove that it is a\nhomological epimorphism,  both for a  finite-dimensional nilpotent\ncomplex Lie algebra $\\fg$.  (Here $U(\\fg)$ denotes the universal\nenveloping algebra of $\\fg$.)\n\n\\subsection*{Arens-Michael envelopes}\nIn this text, all vector spaces and algebras are considered over\nthe field $\\CC$ of complex numbers. All algebras and their\nhomomorphisms are assumed to be unital.\n\nRecall that  a complete Hausdorff locally convex topological\nalgebra with jointly continuous multiplication is called a\n\\emph{$\\ptn$-algebra}. (Here $\\ptn$ is the sign for the complete\nprojective tensor product of locally convex spaces.)  A~$\\ptn$-\nalgebra A is called an \\emph{Arens-Michael algebra} (or a\n\\emph{complete m-convex algebra}) if its topology is determined by\na family of submultiplicative prenorms $(\\|\\cdot\\|_\\al)$ (i.e.,\n$\\|ab\\|_\\al \\le\\|a\\|_\\al\\|b\\|_\\al$ for all $a, b\\in A$).\n\n\nThe \\emph{Arens-Michael envelope} of a $\\ptn$-algebra $A$\n\\cite[Chap.~5]{X2} is  a pair $(\\widehat A, \\io_A)$, where\n$\\widehat A$ is an Arens-Michael algebra  and $\\io_A$ is a\ncontinuous homomorphism $A \\to \\widehat A$ s.t. for any\nArens-Michael algebra $B$ and for each continuous homomorphism\n$\\phi\\!: A \\to B$ there exists a unique continuous homomorphism\n$\\widehat\\phi\\!:\\widehat A \\to B$ making the following diagram\ncommutative\n\\begin{equation}\\label{AMen}\n  \\xymatrix{\nA \\ar[r]^{\\io_A}\\ar[rd]_{\\phi}&\\widehat A\\ar@{-->}[d]^{\\widehat\\phi}\\\\\n &B\\\\\n }\n\\end{equation}\nNote that it suffices to consider only homomorphisms with values\nin Banach algebras. The Arens-Michael envelope always exists and\nis unique up to a topological isomorphism. The algebra $\\widehat\nA$ is the completion of $A$ w.r.t. the family of all continuous\nsubmultiplicative prenorms. (An  arbitrary associative\n$\\CC$-algebra can be considered as a $\\ptn$-algebra w.r.t. the\nstrongest locally convex topology; in this case, all prenorms are\nautomatically continuous.)\n\nTo formulate our first main result recall some terminology and\nnotation from \\cite{Go78,Go79,Pir_stbflat}.  Consider a\nfinite-dimensional nilpotent complex Lie algebra~$\\fg$ and fix a\npositive decreasing filtration $\\mathscr{F}$ on~$\\fg$, i.e., a\ndecreasing chain of subspaces\n\\begin{equation}\\label{posfilt}\n\\fg=\\fg_1 \\supset \\fg_2 \\supset \\cdots  \\supset\\fg_k \\supset\\fg_{k+1}=0,\n\\qquad [\\fg_i,\\fg_j]\\subset \\fg_{i+j}\\,.\n\\end{equation}\nConsider a basis $(e_1,\\ldots, e_m)$ in $\\fg$ and set\n\\begin{equation}\\label{widef}\nw_i\\!:=\\max\\{j:\\,e_i\\in \\fg_j \\}\\quad\\text{and}\\quad\nw(\\alpha)\\!:=\\sum_i w_i\\alpha_i \\,,\n\\end{equation}\nwhere $\\alpha=(\\alpha_1,\\ldots,\\alpha_m)\\in\\Z_+^m$. In the\nfollowing  we assume that $(e_i)$ is  an\n\\emph{$\\mathscr{F}$-basis}, i.e., $w_i\\le w_{i+1}$  for all $i$,\nand $\\fg_j = \\spn\\{e_i : w_i \\ge j\\}$ for all $j$. A sequence\n${\\mathscr M}=\\{ M_\\alpha: \\alpha\\in\\Z_+^m\\}$ of positive numbers\nis called an \\emph{${\\mathscr F}$-weight sequence} if $M_0=1$ and\n$M_\\gamma\\le M_\\alpha M_\\beta$ whenever $w(\\gamma)\\ge\nw(\\alpha)+w(\\beta)$. Given an ${\\mathscr F}$-weight sequence\n${\\mathscr M}$, consider the space\n\\begin{multline}\\label{UfgMde}\nU(\\fg)_{\\mathscr M}\\!:= \\Bigl\\{ x=\\sum_{\\alpha\\in\\Z_+^m} c_\\alpha\ne^\\alpha\\in [U] \\!:\\\\\n \\| x\\|_r\\!:=\\sum_\\alpha |c_\\alpha|\n\\alpha!\\, M_\\alpha r^{w(\\alpha)}<\\infty \\;\\forall r>0\\Bigr\\}\\,,\n\\end{multline}\n where $(e^\\al\\,:\\al\\in\\Z_+^m)$ is the PBW-basis in\n$U(\\fg)$ associated with  $(e_i)$, and $[U]$ is the set of formal\npower series w.r.t. $(e^\\al)$. It is proved in\n\\cite[Th.~6.3]{Go78} that the multiplication on  $U(\\fg)$ extends\nto $U(\\fg)_{\\mathscr M}$ and $U(\\fg)_{\\mathscr M}$ is a\n$\\ptn$-algebra.\n\nThe standard choice for $\\mathscr{F}$ is  the lower central\nseries of $\\fg$ that is defined inductively by $\\fg_i\\!:=[\\fg,\n\\fg_{i-1}]$. Consider the $\\mathscr F$-weight sequence $\\mathscr\nM_1$ defined by $ M_\\al\\!:=w(\\al)^{-w(\\al)}$. (This is $\\mathscr\nM_p$ as defined in \\cite[Sec.~2, Exm.~1]{Go79} with $p=1$.)\n\n\n\\begin{thm}\\label{AMEdescrnil}\nLet $\\fg$ be a finite-dimensional nilpotent complex Lie algebra,\n$\\mathscr{F}$  the lower central series, and $(e_i)$ an\n$\\mathscr{F}$-basis.  Then the topology on the $\\ptn$-algebra\n$$\nU(\\fg)_{\\mathscr M_1}= \\Bigl\\{ x=\\sum_{\\alpha\\in\\Z_+^m} c_\\alpha\ne^\\alpha\\in [U]\\! : \\| x\\|_r=\\sum_\\alpha |c_\\alpha| \\alpha!\\,\nw(\\al)^{-w(\\al)} r^{w(\\alpha)}<\\infty \\;\\forall r>0\\Bigr\\}\n$$\ncan be determined by system of submultiplicative prenorms, i.e.,\nit is an Arens-Michael algebra. Moreover, the natural\nhomomorphism $U(\\fg)\\to U(\\fg)_{\\mathscr M_1}$ is an\nArens-Michael envelope.\n\\end{thm}\n\n\nThe first step of the proof of Theorem~\\ref{AMEdescrnil} is\nreduction $\\widehat U(\\fg)$ to the Arens-Michael envelope of the\nalgebra $\\mathscr{A}(G)$ of analytic functionals on the\ncorresponding simply connected complex Lie group $G$\n(Proposition~\\ref{csicHFG}). Further, we use  the identification\n(obtained by Akbarov in \\cite{Ak08}) between the strong dual\nspace of $\\widehat{\\mathscr{A}}(G)$  and the locally convex space\n$\\cO_{exp}(G)$ of holomorphic functions of exponential type\non~$G$.\n\nThe key technical result is Theorem~\\ref{eelUsi}, which gives\nestimations for growth rate of a word length function. It is not\nparticularly  original: an explicit formulation and a part of a\nproof can be found in \\cite[II.4.17]{DER}, where the statement is\ngiven for a right invariant Riemannian distance. The reasoning is\nessentially contained in the proofs of \\cite[Pr.~IV.5.6 and\nIV.5.7]{VSC92} but in a latent form.  Moreover, the main goal of\n[ibid.] is to rate volume growth; so the reader needs some\nadditional work to extract an argument for distances. Very close\nresults are contained in \\cite[Th.~4.2]{Kar94} and\n\\cite[Pr.~7.25]{Be96} but in variations that are not completely\nsatisfactory for our purposes. Besides, a careful examination\nshows that estimates for Riemannian distances are based on\nestimates for length functions; thus there is a direct way to\nestablish Theorem~\\ref{eelUsi}, which passes Riemannian geometry.\nSo I include a complete proof in Appendix, where a connection\nwith Riemannian distances is also explained.\n\n\nFrom Theorem~\\ref{eelUsi} we obtain an explicit description of\n$\\cO_{exp}(G)$ (Theorem~\\ref{exptypdesc}) and show that the\nfunctions of exponential type forms exactly the dual space of\n$U(\\fg)_{\\mathscr{M}_1}$, which implies our assertion.\n(Theorem~\\ref{exptypdesc}, which plays only supporting role in our\nargument, is of independent interest itself. This result is an\nessential part of the description of the space of holomorphic\nfunctions of exponential type on an arbitrary connected complex\nLie group --- the subject that is discussed in \\cite{Ar19}.)\n\n\\subsection*{Homological epimorphisms}\nLet $A$ be a $\\ptn$-algebra. Recall that an\n\\emph{$A$-$\\ptn$-bimodule} is a complete Hausdorff locally convex\nspace endowed with a structure of a unital  $A$-bimodule s.t. both\nleft and right multiplications are jointly continuous. Below\n$\\ptens{A}$ denotes the projective tensor product of\n$A$-$\\ptn$-modules.\n\nA homomorphism of $\\ptn$-algebras $A\\to B$ is called\na~\\emph{homological epimorphism} if the induced functor between\nthe bounded derived categories of $\\ptn$-modules is fully\nfaithful. This condition is equivalent to the following:  for some\n(or what is the same, for each) admissible projective resolution\n$0 \\lar A \\lar L_\\bullet$ in the relative category of\n$A$-$\\ptn$-bimodules the\n complex\n\\begin{equation}\\label{nqflres}\n0 \\lar B\\ptens{A} B  \\lar B\\ptens{A} L_0\\ptens{A} B   \\lar \\cdots\n \\lar B\\ptens{A} L_{n}\\ptens{A} B  \\lar\\cdots\n\\end{equation}\nis admissible \\cite[Rem.~6.4]{Pir_qfree}.  We follow the\nterminology from [ibid.]; the alternative terminologies: $A\\to B$\nis a \\emph{localization} or $B$ is \\emph{stably flat over $A$} are\nused in \\cite{Pir_stbflat}. Definitions of homological notions for\n$\\ptn$-algebras can be found in \\cite{X1,X2,X_HOA}. We do not need\ndetails here because the only necessary fact on homological\nepimorphisms is Theorem~\\ref{PirkThUM} below.\n\n\nTaylor \\cite{T2} proved that the Arens-Michael envelope is a\nhomological epimorphism for the polynomial algebra in $n$\ngenerators and for the free algebra in $n$ generators.  The\nnatural next step is to consider the universal enveloping algebra\n$U(\\fg)$ of a finite-dimensional complex Lie algebra $\\fg$. The\nfirst and a bit disappointing result in this direction, also due\nto Taylor \\cite{T2}, asserts that if $\\fg$ is semisimple, then the\nArens-Michael envelope $\\io_U\\!:U(\\fg)\\to \\widehat U(\\fg)$ fails\nto be a homological epimorphism, in contrast to the abelian case.\nMany years later the results for solvable $\\fg$ began to appear.\nDosiev \\cite[Th.~10]{Do03} proved that $\\io_U$ is a homological\nepimorphism provided $\\fg$ is metabelian (i.e., $[\\fg, [\\fg, \\fg]]\n= 0$). Later, Pirkovkii generalized this result to positively\ngraded Lie algebras \\cite[Th.~6.19]{Pir_stbflat} and Dosiev\ngeneralized it to nilpotent Lie algebras satisfying a condition of\n''normal growth'' \\cite{Do09}. A natural conjecture became that\nthe same is true for each nilpotent Lie algebra $\\fg$. On the\nother hand, it was shown in \\cite{Pi4} that  $\\io_U$   can be a\nhomological epimorphism only when $\\fg$ is solvable.\n\nAnother approach was introduced Pirkovkii in \\cite{Pir_qfree}. An\nOre extension iteration gives sometimes a direct construction of\n$\\widehat U(\\fg)$  and a method to prove that $\\io_U$ is a\nhomological epimorphism. This method requires some technical work;\nnonetheless, it gives the only known up to date example of a\nsolvable non-nilpotent~$\\fg$ s.t. $\\io_U$ is a homological\nepimorphism, namely, the two-dimensional solvable non-abelian Lie\nalgebra.\n\n\n\nNow we formulate our second main result.\n\\begin{thm}\\label{thmmail}\nLet $\\fg$ be a finite-dimensional nilpotent complex Lie algebra.\nThen the Arens-Michael envelope $\\io_U\\!:U(\\fg)\\to \\widehat\nU(\\fg)$  is a   homological epimorphism.\n\\end{thm}\nThe  proof is based on Theorem~\\ref{AMEdescrnil} and the\nfollowing Pirkovskii's theorem. (The definition of an entire\n${\\mathscr F}$-weight sequence see below in~\\eqref{entire}.)\n\\begin{thm} \\label{PirkThUM}\n\\cite[Th.~7.3]{Pir_stbflat} Let $\\fg$ be a finite dimensional\nnilpotent complex Lie algebra,  and let ${\\mathscr M}$ be an\nentire ${\\mathscr F}$-weight sequence for some positive\nfiltration ${\\mathscr F}$. Then the natural homomorphism\n$U(\\fg)\\to U(\\fg)_{\\mathscr M}$ is a   homological epimorphism.\n\\end{thm}\n\n\nThe present paper is organized as follows. In section~\\ref{PR},\nsome preliminary results on the algebra of analytic functionals,\nsubmultiplicative weights, and holomorphic functions of\nexponential type are collected.  In section~\\ref{RM}, a result on\ngrowth of word length functions (Theorem~\\ref{eelUsi}) is\nformulated  and applied to characterize  holomorphic functions of\nexponential type on a simply connected nilpotent  Lie group\n(Theorem~\\ref{exptypdesc}). The proofs of\nTheorems~\\ref{AMEdescrnil},~\\ref{thmmail}  and some application\nare contained in  section~\\ref{PMR}. Appendix includes the proof\nof Theorem~\\ref{eelUsi} and the explanation how this assertion is\nconnected with growth rate of invariant Riemannian distances.\n\n\\subsection*{Acknowledgements} The author thanks S.~Akbarov for\nuseful comments.\n\n\\section{Preliminary results}\\label{PR}\n\\subsection*{Reduction to the algebra of analytic functionals}\n\nLet $G$ be a complex Lie group with Lie algebra $\\fg$.  Consider\nthe Fr\\'{e}chet algebra $\\cO(G)$ of holomorphic functions on $G$\nand its strong dual space $\\cO(G)'$ endowed with the convolution\nmultiplication.  This $\\ptn$-algebra is denoted by\n${\\mathscr{A}}(G)$ and is called the \\emph{algebra of analytic\nfunctionals} on $G$ (for a general complex Lie group,\n${\\mathscr{A}}(G)$ is introduced by G.~Litvinov in \\cite{Lit}).\n\nConsider elements of $U(\\fg)$ as left-invariant differential\noperators on $\\cO(G)$ and define the homomorphism $\\tau$ by\n\\begin{equation}\\label{taudef}\n\\tau\\!:U(\\fg) \\to \\mathscr{A}(G)\\!:\\langle \\tau(X), f\\rangle\\!:= \\langle\n\\de_e, Xf\\rangle \\qquad (X\\in U(\\fg),\\, f \\in  \\cO(G))\\,,\n\\end{equation}\nwhere $\\de_e$ is denoted the delta-function at $e$.\n\nRecall that the Arens-Michael envelope is a functor from the\ncategory of $\\ptn$-algebras to the category of  Arens-Michael\nalgebras. For any  $\\ptn$-algebra homomorphism $\\te$ we denote\nthe corresponding homomorphism between the Arens-Michael\nenvelopes by $\\widehat \\te$.\n\n\\begin{pr}\\label{csicHFG}\nLet $G$ be a simply connected complex Lie group with Lie\nalgebra~$\\fg$. Then $\\widehat\\tau\\!:\\widehat{U}(\\fg) \\to\n\\widehat{\\mathscr{A}}(G)$ is an Arens-Michael algebra\nisomorphism.\n\\end{pr}\n\\begin{rem}\nThe similar result is valid for real Lie groups.  If $G$ is a\nsimply connected real Lie group and  $\\fg_\\CC$ denotes the\ncomplexification of its Lie algebra, then   $U(\\fg_\\CC)$ and the\nalgebra $\\mathcal{E}'(G)$ of compactly supported distribution\nhave  the  same Arens-Michael envelope  \\cite[p.~250]{T2}.\n\\end{rem}\nTo prove Proposition~\\ref{csicHFG} we need the following lemma.\n\\begin{lm}\\label{BwhA}\nLet $\\te\\!:A\\to B$ be an epimorphism of $\\ptn$-algebras. Suppose\nthat  there exists a continuous homomorphism $j\\!:B\\to\n\\widehat{A}$ s.t. $\\io_{A}=j\\te$.   Then\n$\\widehat{\\te}\\!:\\widehat{A}\\to \\widehat{B}$ is an Arens-Michael\nalgebra isomorphism.\n\\end{lm}\n\\begin{proof}\nThe homomorphisms $\\io_{A}$ and $\\io_{B}$ have dense ranges; so\nthey are epimorphisms. Since, by assumption, $\\te$ is an\nepimorphism, it follows from $\\widehat{\\te}\\io_{A}=\\io_{B} \\te$\nthat  $\\widehat{\\te}$ also is  an epimorphism.\n\nBy the universal the property of the Arens-Michael envelope, there\nis a continuous homomorphism $\\al\\!:\\widehat{B}\\to \\widehat{A}$\ns.t. $j=\\al\\io_{B}$. Therefore,\n$$\n\\al \\widehat{\\te}\\io_{A}=\\al\\io_{B} \\te=j\\te=\\io_{A}\\,.\n$$\nSince $\\io_{A}$  is an epimorphism, we have $\\al\n\\widehat{\\te}=1$.   Therefore, $\\widehat{\\te}\\al\n\\widehat{\\te}=\\widehat{\\te}$. But $\\widehat{\\te}$  is an\nepimorphism so  $\\widehat{\\te}\\al =1$. Thus $\\widehat{\\te}$ is\ninvertible.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{csicHFG}]\nSince $G$ is simply connected, it follows from\n\\cite[Pr.~9.1]{Pir_stbflat} that there exists a unique continuous\nhomomorphism $j\\!:{\\mathscr{A}}(G)\\to \\widehat{U}(\\fg)$ s.t.\n$\\io_{U(\\fg)}=j\\tau$. On the other hand, $\\tau$ has dense range\nprovided  $G$ is connected (because the dual map is injective;\nsee, e.g., discussion after formula~(42) in [ibid.]). So we can\napply Lemma~\\ref{BwhA}.\n\\end{proof}\n\n\n\n\n\\subsection*{Submultiplicative weights and length functions}\n\n\\begin{df}\n(A) A \\emph{submultiplicative weight} on a locally compact group $G$ is a\nnon-negative locally bounded function $\\om\\!: G \\to \\R$ s.t.\n$$\n\\om(gh)\\le \\om(g)\\om(h)\\qquad (g, h \\in G)\\,.\n$$\n\n(B)  A \\emph{length function} on a locally compact group $G$ is a locally\nbounded function $\\ell\\!:G\\to \\R$ s.t.\n$$\n \\ell(gh)\\le \\ell(g)+\\ell(h)\\qquad (g, h \\in G)\\,.\n$$\n\\end{df}\n\nIt is not hard to check that a strictly positive submultiplicative\nweight maps $G$ to $[1,+\\infty)$, and a length function  maps $G$\nto $[0,+\\infty)$.\n\n\\begin{rem}\n(A) We accept the terminology from \\cite{Wi13}. As a rule, $\\ell$\n(or $\\om$) is assumed to be symmetric, i.e., $\\ell(e) = 0$ and\n$\\ell(g^{-1})=\\ell(g)$ (or $\\om(e) = 1$ and $\\om(g^{-1})=\\om(g)$);\nsee Appendix. The short term 'weight' (see, e.g.,\n\\cite{Sch93,Da00}) is usual but it has too many different senses.\nAkbarov in \\cite{Ak08} employs 'semicharacter' for\n'submultiplicative weight'. L.~Schweitzer in \\cite{Sch93} prefers\n'gauge' for 'length function'.  The words  'seminorm' for\n'submultiplicative weight' \\cite[Sect.~4.2.2]{War72} and 'modulus'\nfor 'length function' \\cite{DER} can be also  used in the Lie\ngroup context.\n\n(B) We assume that $\\om$ or $\\ell$ is locally bounded because we\nfollow \\cite{Ak08} principally. The more common admissions that\n$\\om$ or $\\ell$ is measurable (w.r.t. the Haar measure) or Borel\nare stronger. Indeed, in these cases, submultiplicativity or\nsubadditivity   implies that $\\om$ or $\\ell$, resp., is bounded\non compact sets and, hence, is locally bounded (see\n\\cite[Pr.~2.1]{Dz86} and \\cite[Th.~1.2.11]{Sch93}).\n\n(C)  The  map $\\ell\\mapsto (\\om(g)\\!:=e^{\\ell(g)})$ is a\nbijection between the set of length functions and the set of\nsubmultiplicative weights.\n\\end{rem}\n\nWe use the following notation. For a complex manifold $M$ and   a\nlocally bounded function $\\up\\!:M\\to [1,+\\infty)$ denote by\n$V_\\up$  the closed absolutely convex hull of\n$$\n\\{\\up(x)^{-1}\\de_x:\\,x\\in M\\}\n$$\nin $\\mathscr{A}(M)\\!:=\\cO(M)'$. It is noted in\n\\cite[Sect.~3.4.3]{Ak08} that $V_\\up$ is a neighbourhood of~$0$ in\n$\\mathscr{A}(M)$; so $V_\\up$ is an absorbent set. Therefore its\nMinkowski functional is well defined on $\\mathscr{A}(M)$; we\ndenote it by $\\|\\cdot\\|_{\\up}$. Let  ${\\mathscr{A}}_\\up(M)$ be the\ncompletion of ${\\mathscr{A}}(M)$ w.r.t. $\\|\\cdot\\|_{\\up}$. Also,\ndenote by ${\\mathscr{A}}_{\\up^\\infty}(M)$ the completion of\n${\\mathscr{A}}(M)$ w.r.t. the sequence of prenorms\n$(\\|\\cdot\\|_{\\up^n};\\,n\\in\\N)$, where $\\up^n(x)\\!:=\\up(x)^n$.\n\nFor any submultiplicative weight $\\om$ on a complex Lie group $G$\nthe prenorm~$\\|\\cdot\\|_{\\om}$ is  submultiplicative and continuous\non $\\mathscr{A}(G)$ [ibid., Lem.~5.1(a)]. (Note that main results\nin [ibid.] is formulated for  Stein groups but their proofs work\nfor all complex Lie groups.) Thus ${\\mathscr{A}}_\\om(G)$ is a\nunital Banach algebra, and the natural map $\\mathscr{A}(G)\\to\n{\\mathscr{A}}_\\om(G)$ is a continuous homomorphism. Obviously, the\nmaximum of two submultiplicative weights is a submultiplicative\nweight, so we have a directed projective system of unital Banach\nalgebras\n$$\n({\\mathscr{A}}_\\om(G):\\,\\text{$\\om$ is a submultiplicative weight on\n$G$})\n$$ with natural connecting homomorphisms. Note that this\nsystem is not empty because the trivial character $g\\mapsto 1$ is\na submultiplicative weight. The following result is a\nreformulation  of  [ibid., Th.~5.2(a)].\n\n\\begin{thm}\\label{AMesw}\nIf $G$ is a complex Lie group, then the continuous homomorphism\n$$\\mathscr{A}(G)\\to \\varprojlim_\\om  {\\mathscr{A}}_\\om(G)$$ is an\nArens-Michael envelope.\n\\end{thm}\n\n\nLet $U$ be a  generating set for  $G$, i.e., $e\\in U$ and\n$\\bigcup_{n=0}^{\\infty} U^{n} = G$, where $U^{0}\\!:=\\{e\\}$.\nRecall that a locally compact group $G$ is called \\emph{compactly\ngenerated} if there is a relatively compact generating set $U$.\nFor given $U$, we define a  function $\\ell_U$ on $G$ by\n\\begin{equation}\\label{wordlen}\n\\ell_U(g)\\!: = \\min \\{ n \\!: \\, g \\in U^{n} \\}\\,.\n\\end{equation}\nIt is easy to see that $\\ell_U$ is a length function, it is called\na \\emph{word length function} (cf., e.g.,\n\\cite[Exm.~1.1.7]{Sch93}).\n\nFor given non-negative functions  $\\tau_1$ and $\\tau_2$ on a set\n$X$ we say that $\\tau_1$   \\emph{dominated by}  $\\tau_2$ (at\ninfinity) if there are $C,D>0$ s.t.\n$$\\tau_1(x)\\le C\\tau_2(x) + D\\qquad (x\\in X)\\,.$$\nNon-negative functions  $\\tau_1$ and $\\tau_2$ on a set $X$ are\nsaid to be \\emph{equivalent} (at infinity) if $\\tau_1$ dominated\nby $\\tau_2$   and $\\tau_2$   dominated by $\\tau_1$.\n\n\\begin{pr} \\label{decprli}\n\\emph{(cf. \\cite[Th.~5.3]{Ak08})} Let $G$ be a compactly generated\ncomplex Lie group, and let $U$ be a relatively compact generating\nset. Put\n$$\n\\xi(g)\\!:=e^{\\ell_U(g)} \\,,\n$$\nwhere $\\ell_U$ is the word length function defined by~\\eqref{wordlen}. Then\nthe sequence of submultiplicative prenorms $(\\|\\cdot\\|_{\\xi^n};\\,n\\in \\N)$\ndetermines the topology on  $\\widehat{\\mathscr{A}}(G)$, i.e.,\n$$\n\\widehat{\\mathscr{A}}(G)\\cong  {\\mathscr{A}}_{\\xi^\\infty}(G)\\,.$$\n\\end{pr}\n\\begin{proof}\nIt is easy to show (see \\cite[Th.~1.1.21]{Sch93} or\n\\cite[Th.~5.3]{Ak08}) that every length function on $G$ is\ndominated  by $\\ell_U$.  Therefore for each submultiplicative\nweight $\\om$ on $G$ there are $C>0$ and $n\\in\\N$ s.t.\n$\\|\\cdot\\|_\\om\\le C\\|\\cdot\\|_{\\xi^n}$.\n\\end{proof}\n\n\\subsection*{Holomorphic functions of exponential type}\n\nFor  a complex Lie group $G$, denote by $\\cO_{exp}(G)$ the linear\nsubspace of $\\cO(G)$ that contains every function $f$ s.t. there\nis a submultiplicative weight $\\om$ satisfying  $|f(g)|\\le \\om(g)$\nfor all $g\\in G $. A~holomorphic function $f$ on $G$ is of\n\\emph{exponential type}, if  $f\\in\\cO_{exp}(G)$\n\\cite[Sect.~5.3.1]{Ak08}.\n\nTo define the topology on $\\cO_{exp}(G)$ we use the following notation. For\na complex manifold $M$ and  a locally bounded function $\\up\\!:M\\to\n[1,+\\infty)$ denote by $\\cO_\\up(M)$ the linear subspace of $\\cO(M)$ defined\nby\n\\begin{equation} \\label{fupn}\n\\cO_\\up(M)\\!:=\\Bigl\\{ f\\in\\cO(M) \\!: |f|_\\up\\!:=\\sup_{x\\in\nM}{\\up(x)}^{-1}{|f(x)|}<\\infty\\Bigr\\}\\,.\n\\end{equation}\nIt is easy to see that  $\\cO_\\up(M)$ is a Banach space w.r.t\n$|\\cdot|_\\up$.  Put $\\cO_{\\up^\\infty}(M)\\!:=\\bigcup_{n\\in\\N}\n\\cO_{\\up^n}(M)$. We consider $\\cO_{\\up^\\infty}(M)$ with the\ninductive limit topology.\n\nFor a complex Lie group $G$, we have an inductive system of\nBanach spaces\n$$(\\cO_\\om(G):\\,\\text{$\\om$ is a submultiplicative weight on\n$G$})$$ with natural connecting homomorphisms. Note that\n$\\cO_{exp}(G)=\\bigcup_{\\om} \\cO_\\om(G)$. So we can consider $\\cO_{exp}(G)$\nas a locally convex space via  identification\n$$\n\\cO_{exp}(G)=\\varinjlim_{\\om} \\cO_\\om(G).\n$$\n\n\n\n\\begin{pr} \\label{decinli}\\emph{(cf. \\cite[Th.~5.3]{Ak08})}\nLet $G$ be a compactly generated complex Lie group,  $U$  a\nrelatively compact generating set, and $\\xi$  defined as in\nProposition~\\ref{decprli}. Then\n$$\n\\cO_{exp}(G)=  \\cO_{\\xi^\\infty}(G)\n$$\nas locally convex spaces.\n\\end{pr}\n\\begin{proof}\nNote again that  each length function on $G$ is dominated  by $\\ell_U$.\n\\end{proof}\n \\begin{rem}\nIn~\\cite{Ak08}, Akbarov introduces  $\\cO_{exp}(G)$  as an\ninductive limit of the system $(\\cO_{\\om}(G))$, where each\n$\\cO_{\\om}(G)$ is endowed with the topology of a Smith space, and\nincludes his results in more general context. In particular, if\n$E$  is a stereotype locally convex space, he considers the dual\nspace $E^\\bigstar$ endowed the topology of uniform convergence on\ntotally bounded subsets. Since each relatively compact subset of a\ncomplete metric space is totally bounded  and each bounded subset\nof a nuclear space is relatively compact \\cite[\\S~III.7.2,\nCor.~2]{SM}, we have $E^\\bigstar=E'$ for every nuclear Fr\\'echet\nspace $E$. So far as $\\cO_{exp}(G)$ is dual to the nuclear\nFr\\'echet space $\\widehat{\\mathscr{A}}(G)$ (see\nProposition~\\ref{AOdul} below), our and Akbarov's approaches to\ntopology are equivalent. In addition, note that there is an\nalternative proof of Proposition~\\ref{AOdul}, which based\non~\\cite[Th.1.11]{Ak08}.\n\\end{rem}\n\n\n\n\\begin{lm}\\label{AupOup}\nLet $M$ be a complex manifold.  For given locally bounded\nfunction  $\\up\\!:M\\to [1,+\\infty)$, the pairing between\n${\\mathscr{A}}(M)$  and $\\cO(M)$  induces the pairing  that makes\n$\\cO_\\up(M)$ the dual Banach space to ${\\mathscr{A}}_\\up(M)$.\n\\end{lm}\n\\begin{proof}\nDenote by $\\langle \\cdot,\\cdot\\rangle$ the pairing between\n${\\mathscr{A}}(M)$  and $\\cO(M)$. Evidently,\n$$S_\\up\\!:=\\{f\\in \\cO(M):\\,|f(x)|\\le \\up(x)\\, \\forall x \\in M\\}$$\nis the unit ball in  $\\cO_\\up(M)$.   By \\cite[Lem.~3.1]{Ak08}, the\nset $S_\\up$ is the polar of $V_\\up$ (the closed  absolutely convex\nhull of $\\{\\up(x)^{-1}\\de_x:\\,x\\in M\\}$) in ${\\mathscr{A}}(M)$.\nTherefore,\n$$\n|\\langle \\mu,f \\rangle|\\le \\|\\mu\\|_\\up\\,|f|_\\up\\qquad (\\mu\\in\n{\\mathscr{A}}_\\up(M),\\,f\\in\\cO_\\up(M))\\,.\n$$\nSo we have a bounded linear operator $\\al\\!:\\cO(M)_\\up\\to\n{\\mathscr{A}}_\\up(M)'$.\n\nOn the other hand,  suppose that $h\\in {\\mathscr{A}}_\\up(M)'$. Let\n$\\rho_\\up\\!:{\\mathscr{A}}(M)\\to {\\mathscr{A}}_\\up(M)$ be the\ncompletion map.  Then $h\\rho_\\up\\in {\\mathscr{A}}(M)'$ and the\ncorresponding function in $\\cO(M)$ is determined by $x\\mapsto\nh\\rho_\\up(\\de_x)$. Denote this function by $f(x)$. Since $h$ is\nbounded,  there is $C>0$ s.t. $|f(x)|\\le C\\|\\de_x\\|_\\up\\le\nC\\up(x)$ for all $x\\in M$, i.e., $f\\in \\cO_\\up(M)$ and $|f|_\\up\\le\nC$.   So we have a bounded linear operator\n${\\mathscr{A}}_\\up(M)'\\to\\cO(M)_\\up $, which we denote by $\\be$.\n\nIt is obvious that $\\be\\al=1$. By the definition of\n$\\|\\cdot\\|_\\up$, the linear span of $\\{\\de_x:\\,x\\in M\\}$ is dense\nin ${\\mathscr{A}}_\\up(M)$, so the operator $\\be$ is injective.\nTherefore we have a topological isomorphism.\n\\end{proof}\n\n\\begin{lm} \\label{AOdul0}\nLet $M$ be a complex manifold.  For given locally bounded\nfunction  $\\up\\!:M\\to [1,+\\infty)$ and any  $n\\in\\N$ consider the\npairing $\\langle \\cdot,\\cdot \\rangle_{\\up^n}$ between\n${\\mathscr{A}}(M)_{\\up^n}$  and $\\cO(M)_{\\up^n}$ from\nLemma~\\ref{AupOup}. If the Fr\\'{e}chet space\n$\\mathscr{A}_{\\up^\\infty}(M)$ is reflexive, then there is a\nparing between $\\mathscr{A}_{\\up^\\infty}(M)$  and\n$\\cO_{\\up^\\infty}(M)$ that is compatible with all $\\langle\n\\cdot,\\cdot \\rangle_{\\up^n}$ and making\n${\\mathscr{A}}(M)_{\\up^\\infty}$  and $\\cO(M)_{\\up^\\infty}$ strong\ndual spaces to each other.\n\\end{lm}\n\\begin{proof}\nEvery reflexive Fr\\'{e}chet space is distinguished, i.e., the\nstrong  dual is barreled \\cite[\\S 23.7]{Kot1}. For every\nrepresentation $E = \\varprojlim E_n$ of a distinguished\nFr\\'{e}chet space $E$ as a reduced (each $E\\to E_n$ has dense\nrange) projective limit of a sequence of Banach spaces, its strong\ndual space $E'$ equals $\\varinjlim E_n'$ (see, e.g.,\n \\cite{BDi}, Introduction).    So Lemma~\\ref{AupOup} implies that\n$\\mathscr{A}_{\\up^\\infty}(G)'\\cong \\cO_{\\up^\\infty}(G)$. Finally,\nreflexivity implies  $\\cO_{\\up^\\infty}(G)'\\cong\n\\mathscr{A}_{\\up^\\infty}(G)$.\n\\end{proof}\n\n\\begin{pr} \\label{AOdul}\nLet $G$ be a compactly generated complex Lie group.  Then there is a paring\nbetween    $\\widehat{\\mathscr{A}}(G)$  and $\\cO_{exp}(G)$ that is compatible\nwith all $\\langle \\cdot,\\cdot \\rangle_\\om$, where $\\om$ is a\nsubmultiplicative weight, and making  $\\widehat{\\mathscr{A}}(G)$  and\n$\\cO_{exp}(G)$ strong dual spaces to each other.\n\\end{pr}\n\\begin{proof}\nIt follows from Propositions~\\ref{decprli}  and~\\ref{decinli} that\n$$\n\\widehat{\\mathscr{A}}(G)\\cong\n{\\mathscr{A}}_{\\xi^\\infty}(G)\\quad{\\text{and}}\\quad \\cO_{exp}(G)\\cong\n\\cO_{\\xi^\\infty}(G)\\,.\n$$\nBy \\cite[Ths.~5.10, 6.2]{Ak08}, $\\widehat{\\mathscr{A}}(G)$ is\nnuclear. Therefore it is reflexive and we can apply\nLemma~\\ref{AOdul0}.\n\\end{proof}\n\n\nDenote by ${\\mathscr P}(G)$ the algebra of polynomial functions\non a simply connected complex Lie group $G $ w.r.t. the canonical\ncoordinates of the first kind (i.e., via the identification\n$\\exp\\!:\\fg\\to G$) and consider the natural\nparing\n\\begin{equation}\\label{ufPp}\n \\langle X,f\\rangle\\!:= Xf(e)\\qquad (X\\in U(\\fg),\\,\nf\\in {\\mathscr P}(G)).\n\\end{equation}\n\n\n\\begin{co} \\label{paUwhOe}\nLet $G$ be a simply connected complex Lie group with Lie\nalgebra~$\\fg$. Then the locally convex spaces $\\widehat{U}(\\fg)$\nand $\\cO_{exp}(G)$ are strong dual spaces to each other w.r.t.\nthe pairing extending the pairing between $U(\\fg)$ and ${\\mathscr\nP}(G)$.\n\\end{co}\n\\begin{proof}\nObviously $\\langle X,f\\rangle= \\langle \\de_e, Xf\\rangle$, so the\nhomomorphism $\\tau\\!:U(\\fg) \\to \\mathscr{A}(G)$ defined\nin~\\eqref{taudef} induces a pairing between $U(\\fg)$ and $\\cO(G)$\nthat is continuously extended to $\\widehat{U}(\\fg)$. Finally, note\nthat each simply connected complex Lie group is compactly\ngenerated \\cite[Th.~7.4]{HeRo} and apply\nPropositions~\\ref{csicHFG} and~\\ref{AOdul}.\n\\end{proof}\n\n\\section{Growth of  word length functions}\\label{RM}\nLet $\\fg$ be a nilpotent complex or real Lie algebra and\n$\\mathscr{F}$  the lower central series of~$\\fg$, which is\ndefined  by $\\fg_i\\!:=[\\fg, \\fg_{i-1}]$.   Fix an\n$\\mathscr{F}$-basis $e_1,\\ldots, e_m$  in~$\\fg$. For the simply\nconnected Lie group $G$  associated with~$\\fg$ consider the\ncanonical coordinates of the first kind  $(t_1,\\ldots, t_ m)$\nand the canonical coordinates  of the second kind\n$(\\bar{t}_1,\\ldots, \\bar{t}_ m)$ determined by~$(e_i)$, i.e.,\n$$\ng=\\exp\\Bigl(\\sum_{i=1}^m  t_i e_i\\Bigr)=\\prod_{i=1}^m \\exp(\\bar\nt_i\\,e_i) \\qquad(g\\in G)\\,.\n$$\nRemind the definition of $w_i$ from~\\eqref{widef} and set\n\\begin{equation}\n\\label{sidef}\n \\si(g):=\\max_i|t_i|^{1\/w_i}\\,,\\qquad \\bar\\si(g):=\\max_i|\\bar{t}_i|^{1\/w_i}\\,.\n\\end{equation}\nIn  \\cite{Go79} and \\cite{Pir_stbflat} $\\si$ is denoted by\n$|\\cdot|$ and is called 'homogeneous norm'.\n\nThe following theorem is the heart of our argument. See the proof\nin Appendix.\n\n\n\n\n\\begin{thm}\\label{eelUsi}\nLet $G$ be a simply connected nilpotent (complex or real) Lie\ngroup, and let $\\ell$ be a word length function corresponding to\na relatively compact generating set. Then $\\ell$, $\\si$ and\n$\\bar\\si$  are equivalent (at infinity).\n\\end{thm}\n\nNow we can get the following explicit description of\n$\\cO_{exp}(G)$.\n\n\\begin{thm} \\label{exptypdesc}\nLet $G$ be a simply connected nilpotent complex Lie group with\nLie algebra $\\fg$, and let  $(t_1,\\ldots, t_ m)$ be the canonical\ncoordinates of the first kind associated with an\n$\\mathscr{F}$-basis in $\\fg$, where $\\mathscr{F}$ is the lower\ncentral series. Then\n\\begin{multline*}\n\\cO_{exp}(G)= \\bigl\\{f\\in \\cO(G):\\, \\\\ \\exists C>0,\\, \\exists\nr\\in \\R_+ \\, \\text{s.t.}\\,|f(t_1,\\ldots, t_ m)|\\le C\ne^{r\\max_i|t_i|^{1\/w_i}}\\,\\forall t_1,\\ldots, t_ m \\bigr\\}\n\\end{multline*}\nand  we have\n$$\n\\cO_{exp}(G)\\cong \\varinjlim_{r\\in \\R_+} \\cO_{\\eta^r}(G)\n$$\nas locally convex spaces, where $\\eta(t_1,\\ldots, t_\nm)\\!:=e^{\\max_i|t_i|^{1\/w_i}}$ and the Banach space $\\cO_{\\eta^r}(G)$ is\ndefined  as in~\\eqref{fupn}.\n\nMoreover, we can replace $(t_1,\\ldots, t_ m)$ by the canonical\ncoordinates of the second kind.\n\\end{thm}\nWe need the following lemma.\n\\begin{lm}\\label{eqindlim}\nLet $\\tau_1$ and $\\tau_2$ be non-negative locally bounded\nfunctions on a complex manifold~$M$. If $\\tau_1$ and $\\tau_2$ are\nequivalent  and  $\\up_i(x)\\!:=e^{\\tau_i(x)}$ $(i=1,2)$, then\n$\\cO_{\\up_1^\\infty}(M) = \\cO_{\\up_2^\\infty}(M)$ as subset of\n$\\cO(M)$ and as a locally convex space.\n\\end{lm}\n\\begin{proof}\nFor each $p\\in \\N$ there is $q\\in \\N$ s.t.\n$\\cO_{\\up_1^q}(M)\\subset \\cO_{\\up_2^p}(M)$ and this inclusion is a\n continuous linear map. Therefore we have a continuous linear map\n$\\cO_{\\up_1^\\infty}(M)\\to\\cO_{\\up_2^\\infty}(M)$ of inductive\nlimits. Similarly, we get a continuous linear map in the reverse\ndirection. Evidently, these maps are inverse to each other and\nthis completes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{exptypdesc}]\nNote that $G$ is compactly generated and fix a relatively compact\ngenerating set~$U$. Proposition~\\ref{decinli} implies that\n$\\cO_{exp}(G)= \\cO_{\\xi^\\infty}(G)$, where $\\xi(g)=e^{\\ell_U(g)}$.\nEvidently, $ \\eta^r(g)\\!:=e^{r\\si(g)}$ ($r\\in\\R_+$). It follows\nfrom Theorem~\\ref{eelUsi} that $\\si$  is equivalent to $\\ell_U$;\ntherefore $\\cO_{\\xi^\\infty}(M) = \\cO_{\\eta^\\infty}(M)$ by\nLemma~\\ref{eqindlim}. Obviously,\n$$\n\\varinjlim_{r\\in \\R_+} \\cO_{\\eta^r}(G)=\\varinjlim_{n\\in\\N}\n\\cO_{\\eta^n}(G)\\,,\n$$\nso the statement for $\\si$ is proved.\n\nExactly the same argument is applied to $\\bar \\si$.\n\\end{proof}\n\n\\begin{rem}\nIn terms of the classical entire functions theory,\nTheorem~\\ref{exptypdesc} says, in particular, that $\\cO_{exp}(G)$\nconsists of entire functions in $t_1,\\ldots,t_m$ that are of at\nmost order $1\/w_i$ and finite type w.r.t. $t_i$ for all\n$i=1,\\ldots,m$.\n\\end{rem}\n\n\\begin{exm} \\cite[Sect.~5.4.2]{Ak08}\nLet $\\fg$ be an abelian  Lie algebra $\\fg$ with basis\n$e_1,\\ldots,e_m$. Then $w_1=\\cdots=w_m=1$ and\n$\\si(g)=\\max\\{|t_1|,\\ldots,|t_m|\\}$.  So $\\cO_{exp}(G)$ coincides\nwith the set of entire functions  of exponential type (=of at most\norder $1$ and finite type) on $\\CC^m$ as it is defined in the\nclassical theory of several complex variables \\cite[Def.~1.8]{LG}.\nThis example justifies the terminology.\n\\end{exm}\n\n\\begin{exm}\nConsider the $3$-dimensional complex Heisenberg Lie algebra $\\fg$ with basis\n$e_1,e_2,e_2$  and relation $[e_1,e_2]=e_3$.  The lower central series  has\nthe form\n$$\n\\fg=\\fg_1 \\supset\\fg_2=\\spn\\{e_3\\} \\supset\\fg_3=0\\,.\n$$\nThen $w_1=w_2=1$, $w_3=2$, and\n$\\si(g)=\\max\\bigl\\{|t_1|,\\,|t_2|,\\,|t_3|^{1\/2}\\bigr\\}$.\n\\end{exm}\n\n \\begin{exm}\n Let $\\fg$ be the $7$-dimensional Lie algebra with\nbasis $e_1,\\ldots ,e_7$ and commutation relations\n\\begin{gather*}\n[e_1,e_i]=e_{i+1}\\quad (i=2,\\ldots ,6),\\\\\n[e_2,e_3]=-e_6,\\; [e_3,e_4]=e_7,\\; [e_2,e_4]=[e_2,e_5]=-e_7\\,,\n\\end{gather*}\nthe  undefined brackets being zero (see \\cite{Favre} or\n\\cite[Ch.~2, Exm.~III.3(i)]{GoKh}).  Then\n$$\nw_1=w_2=1,\\; w_3=2,\\; w_4=3, \\; w_5=4,\\;w_6=5,\\; w_7=6\\,,\n$$ and\n$$\n\\si(g)=\\max\\bigl\\{|t_1|,\\,|t_2|,\\,|t_3|^{1\/2},\\,|t_4|^{1\/3},\\,|t_5|^{1\/4},\\,|t_6|^{1\/5},\\,|t_7|^{1\/6}\\bigr\\}\\,.\n$$\n\nThis algebra is exhibited  in \\cite[Rem.~6.6]{Pir_stbflat} as an\nexample of a nilpotent Lie algebra that is not contractible and,\nhence, is not positively graded. Thus, it does not satisfy\nconditions of [ibid., Th.~6.19], which asserts that\n$\\io\\!:U(\\fg)\\to \\widehat U(\\fg)$  is a   homological epimorphism\nfor any positively graded Lie algebra $\\fg$. So our\nTheorem~\\ref{thmmail} is stronger that this Pirkovskii's result.\n\\end{exm}\n\n\\begin{rem}\nSince we choose as $\\mathscr{F}$  the lower central series\nof~$\\fg$, the dimensions of $\\fg_1,\\ldots,\\fg_k$ are invariants of\na nilpotent Lie algebra~$\\fg$. Therefore the sequence\n$w_1,\\ldots,w_m$ is also an invariant of~$\\fg$. Of course, this\ninvariant (as well as the isomorphism class of the $\\ptn$-algebra\n$\\cO_{exp}(G)$) is far from sufficient to distinguish  nilpotent\nLie algebras.  To do this one need the comultiplication  on\n$\\cO_{exp}(G)$ inherited from the multiplication on $U(\\fg)$.\n\\end{rem}\n\n\\section{Proofs of  main results and an application}\\label{PMR}\n\nWith Theorem~\\ref{exptypdesc} under arms we can prove\nTheorems~\\ref{AMEdescrnil} and~\\ref{thmmail}.\n\nWe need the following definition \\cite[Def.~2.1]{Go79}: an\n$\\mathscr F$-weight sequence $\\mathscr{M}$  is  \\emph{entire} if\nthe following two conditions are satisfied\n\\begin{gather}\n\\label{entire}\n\\sum_\\alpha M_\\alpha r^{w(\\alpha)} <\\infty\\quad\\text{for all } r>0\\,;\\\\\n\\sup_{\\alpha,\\beta\\ne 0} \\bigl\\{ A^{w(\\alpha)\/w(\\beta)}\nM_\\beta^{1\/w(\\beta)} M_\\alpha^{-1\/w(\\alpha)}\\bigr\\} <\\infty\n\\quad\\text{for some } A>0\\,.\\notag\n\\end{gather}\n\n\\begin{proof}[Proof of Theorem~\\ref{AMEdescrnil}]\nLet $G$ be a simply connected nilpotent complex Lie group with Lie\nalgebra $\\fg$. Given $z\\in\\CC$, define a linear map\n$\\delta_z\\!:\\fg\\to\\fg$ by $\\delta_z(e_i)=z^{w_i}e_i$. We use the\nsame symbol $\\delta_z$ to denote the corresponding holomorphic\nendomorphism of $G$ satisfying $\\delta_z\\circ\\exp=\n\\exp\\circ\\delta_z$.\n\nRecall that our choice of an $\\mathscr F$-weight sequence is\n$\\mathscr M_1$ defined by $ M_\\al\\!:=w(\\al)^{-w(\\al)}$. Consider\nthe growth  function $\\Phi$ associated with $\\mathscr{M}_1$ given\nby\n$$\n\\Phi(g)\\!:=\\sum_\\alpha M_\\alpha \\si(g)^{w(\\alpha)}=\\sum_\\alpha\n\\left(\\frac{\\si(g)}{w(\\al)}\\right)^{w(\\alpha)}\\,,\n$$\nwhere the function $\\si$ is defined in~\\eqref{sidef}.   (Since\n$\\mathscr{M}_1$ is entire \\cite[Sect.~2, Exm.~1]{Go79}, the\nfunction $\\Phi$ is well defined.) Denote by\n$\\cO_{\\mathscr{M}_1}(G)$ the linear subspace of $\\cO(G)$ that\ncontains every function $f$ s.t. there are $C>0$ and $r>0$ and\n$|f(g)|\\le C\\Phi(\\de_r g)$ is satisfied for all $g\\in G $. To\nmake $\\cO_{\\mathscr{M}_1}(G)$ a locally convex space consider,\nfor $r>0$,  the space\n$$\n\\cO_{{\\mathscr{M}_1},r}(G)\\!:=\\Bigl\\{ f\\in\\cO(G) : N_r(f)\\!:=\\sup_{g\\in\nG}\\Phi(\\de_r g)^{-1}|f(g)|<\\infty\\Bigr\\}\\,.\n$$\nEvidently, $\\cO_{{\\mathscr{M}_1},r}(G)$ is a Banach space w.r.t.\nthe norm $N_r$. Since $\\Phi(\\delta_s g)\\le \\Phi(\\delta_r g)$\nwhenever $0\\le s\\le r$, we have $\\cO_{{\\mathscr{M}_1},s}(G)\\subset\n\\cO_{{\\mathscr{M}_1},r}(G)$ for each $s\\le r$, and $N_r(f)\\le\nN_s(f)$ for each $f\\in \\cO_{{\\mathscr M}_1,s}(G)$. Therefore one\nmay identify $\\cO_{\\mathscr{M}_1}(G)$ with the locally convex\nspace $ \\displaystyle\\varinjlim_{s\\in\\R_+}\n\\cO_{{\\mathscr{M}_1},s}(G)$.\n\n\nOn the other hand,  by Theorem~\\ref{exptypdesc},\n$\\cO_{exp}(G)\\cong \\displaystyle{\\varinjlim_{r\\in\\R_+}\n\\cO_{\\eta^r}(G)} $, where $\\eta(g)\\!:=e^{\\si(g)}$. Goodman noted\nin [ibid., (2.6)] that there are positive constants $c,a,C,A$ s.t.\n$$\nce^{a\\si(g)}\\le \\Phi(g) \\le Ce^{A\\si(g)}\\qquad(g\\in G)\\,.\n$$\nApplying the same argument as in the proof of Lemma~\\ref{eqindlim}\nto the inductive limits $ \\displaystyle\\varinjlim_{s\\in\\R_+}\n\\cO_{{\\mathscr{M}_1},s}(G)$ and\n$\\displaystyle{\\varinjlim_{r\\in\\R_+} \\cO_{\\eta^r}(G)}$ we have\nfrom this observation that $\\cO_{\\mathscr{M}_1}(G)=\\cO_{exp}(G)$\nas a set and as a locally convex space.\n\nIt is shown in [ibid., Th.~3.1] that  $U(\\fg)_{\\mathscr{M}_1}$ is\nthe strong  dual  space of $\\cO_{\\mathscr{M}_1}(G)$ via the\npairing that extends the pairing between $U(\\fg)$ and ${\\mathscr\nP}(G)$ defined in~\\eqref{ufPp}\\footnote{Note that to prove this\nresult Goodman used an alternative definition of\n$U(\\fg)_{\\mathscr{M}}$ for a $\\mathscr F$-weight sequence. In\ncontrast to \\eqref{UfgMde}, which introduced in~\\cite{Go78}, he\nconsidered in~\\cite{Go79} series in symmetrization of the\nPBW-basis $(e^\\al\\,:\\al\\in\\Z_+^m)$. Nevertheless, it follows from\n\\cite[Lem,~6.2 and Rem.~(1) on p.~203]{Go78}  that the approaches\nare equivalent.}. On the other hand, by Corollary~\\ref{paUwhOe},\n$\\widehat{U}(\\fg)$ is the strong  dual  space of  $\\cO_{exp}(G)$\nvia the same paring. Thus $U(\\fg)_{\\mathscr{M}_1}$ and\n$\\widehat{U}(\\fg)$  are isomorphic as $\\ptn$-algebras, and\n$U(\\fg)\\to U(\\fg)_{\\mathscr{M}_1}$ is an Arens-Michael envelope.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thmmail}]\nSince $\\mathscr{M}_1$ is entire \\cite[Sect.~2, Exm.~1]{Go79}, we\ncan refer to Pirkovskii's Theorem~\\ref{PirkThUM}, which asserts\nthat, in this case, the natural map  $U(\\fg)\\to U(\\fg)_{\\mathscr\nM_1}$ is a   homological epimorphism.  Thus, the assertion\nfollows Theorem~\\ref{AMEdescrnil}.\n\\end{proof}\n\n\n\n\n\nAs an application of Theorem~\\ref{AMEdescrnil} we obtain an\nestimation of a submultiplicative norm for powers of elements in\na nilpotent complex Lie algebra~$\\fg$. Let $ \\mathscr{F}$ denote,\nas usual, the lower central series.   Given $X\\in\\fg$ s.t. $X\\ne\n0$, set $w(X)\\!:=\\max\\{ j: X\\in\\fg_j\\}$.\n\nIf $\\|\\cdot\\|$ is a submultiplicative prenorm on $U(\\fg)$, then\ndenote by $A$ the completion of $U(\\fg)$ w.r.t. $\\|\\cdot\\|$ and\nby $\\la\\!:\\fg\\to A$ the corresponding Lie algebra homomorphism.\nIt is easy to see from the spectral properties of  Banach\nalgebras that, for any $X\\in [\\fg,\\fg]$, the element $\\la(X)$  is\ntopologically nilpotent, i.e., $\\|\\la(X)^n\\|^{1\/n}=o(1)$. (In\nfact, $\\la([\\fg,\\fg])$ is contained in the Jacobson radical of\n$A$~\\cite{Tu84}.)   Theorem~\\ref{AMEdescrnil} gives us  the\nfollowing decay  estimation.\n\n\\begin{pr}\nLet $A$ be a unital Banach algebra with a submultiplicative norm\n$\\|\\cdot\\|$, let $\\fg$ be a nilpotent complex Lie algebra with\nlower central series $\\mathscr{F}$ as a positive filtration, and\nlet $\\la\\!:\\fg\\to A$ be a Lie algebra homomorphism. Then for each\n$X\\in\\fg\\setminus\\{0\\}$,\n$$\n\\|\\la(X)^n\\|^{1\/n}=O\\biggl(\\frac{1}{n^{w(X)-1}}\\biggr)\\qquad\n(n\\in \\N)\\,.\n$$\n\\end{pr}\nNote that in the case when $w(X)=1$ we have the trivial assertion\nthat $\\|\\la(X)^n\\|^{1\/n}$ is bounded but for $w(X)>1$ the\nstatement is more interesting.\n\n\\begin{proof}\nConsider the system of prenorms $(\\|\\cdot\\|_r;\\, r>0)$ on\n$U(\\fg)$ from Theorem~\\ref{AMEdescrnil}. By the universal\nproperty, on can extend $\\la$ to a homomorphism $U(\\fg)\\to A$.\nThen $\\|\\la(\\cdot)\\|$ is a submultiplicative prenorm on $U(\\fg)$.\nIt follows from Theorem~\\ref{AMEdescrnil} that $\\|\\la(\\cdot)\\|$\nis continuous w.r.t. $(\\|\\cdot\\|_r;\\, r>0)$. Thus it is\nsufficient to show  that $\\|X^n\\|_r^{1\/n}=O(n^{1-w(X)})$  for\nevery $X\\ne0$ and every $r>0$.\n\nFix $X\\in\\fg\\setminus\\{0\\}$. We claim that there is  an\n$\\mathscr{F}$-basis  $(e_i)$ in $\\fg$ s.t. $X=e_i$ for some~$i$.\nIndeed, let $j\\!:=w(X)$. Then $X\\in \\fg_j$ and $X\\not\\in\n\\fg_{j+1}$. Consider an arbitrary $\\mathscr{F}$-basis  $(e_i)$ in\n$\\fg$. Since       $\\fg_j = \\spn\\{e_i : w_i \\ge j\\}$, we have\n$X=\\sum_{w_i \\ge j}c_ie_i$ for some $c_i\\in\\CC$ s.t. there is $i$\nwith $w_i = j$ and $c_i\\ne 0$. So we can replace $e_i$ by $X$ and\nget an $\\mathscr{F}$-basis again.\n\nIf $\\al\\!:=(0,\\ldots,n,\\ldots)$, where $n$ is on the $i$th place, then\n$w(\\alpha)=nw_i$ and $\\al!=n!$. Therefore,\n$$\n\\|e_i^n\\|_r=n!\\left(\\frac{r}{nw_i}\\right)^{nw_i}\\,.\n$$\nHence, $\\|e_i^n\\|^{1\/n}=O(n^{1-w_i})$. Finally, remind that\n$e_i=X$ with $w_i=w(X)$.\n\\end{proof}\n\n\n\\section{Appendix. Proof of Theorem~\\ref{eelUsi} and relations with Riemannian distances}\\label{AppC1}\n\nThe case of Theorem~\\ref{eelUsi} when $\\fg$ is complex  is easily\nreduced for the real case, so below  we suppose that $\\fg$ is a\nnilpotent real Lie algebra.\n\nThe argument consists of three parts and we need three lemmas.\nBoth statements of the first lemma are partial cases of\n\\cite[Lem.~IV.5.1]{VSC92}.\n\\begin{lm}\\label{fromV}\n\\emph{(A)} For each $n\\in\\N$  there are $N\\in\\N$, $s_1,\\ldots,\ns_{N}\\in \\{1,\\ldots,n\\}$, and $\\al_1,\\ldots, \\al_{N}\\in \\R$ s.t.\nfor arbitrary $Y_1,\\ldots, Y_n\\in\\fg$\n\\begin{equation}\\label{exppro}\n\\exp\\Bigl(\\sum_{s=1}^n Y_s\\Bigr)=\\prod_{r=1}^N \\exp(\\al_r\nY_{s_r})\\,.\n\\end{equation}\n\n\\emph{(B)} For each $n\\in\\N$ and each word $U=(u_1,\\ldots,u_j)$\nin $\\{1,\\ldots,n\\}$ there are $N'\\in\\N$, $s'_1,\\ldots, s'_{N'}\\in\n\\{1,\\ldots,n\\}$, and $\\al'_1,\\ldots, \\al'_{N'}\\in \\R$ s.t. for\narbitrary $Y_1,\\ldots, Y_n\\in\\fg$\n\\begin{equation}\\label{exppro2}\n\\exp(Y_U)=\\prod_{p=1}^{N'} \\exp(\\al'_p Y_{s'_p})\\,,\n\\end{equation}\nwhere  $Y_U\\!:=[Y_{u_1},[Y_{u_2},\\cdots, Y_{u_j}]\\cdots]$.\n\\end{lm}\n\n\n\\begin{lm}\\label{sisubpo}\nThere are $C,D\\ge 0$ s.t.\n\\begin{equation}\\label{sig12}\n\\si(g_1 g_2)\\le C\\max\\{\\si(g_1),\\,\\si(g_2)\\}+D\\qquad(g_1,g_2\\in G)\n\\end{equation}\ni.e., $e^\\si$ is 'sub-polynomial' in the terminology of\n\\cite[(1.3.1)]{Sch93}.\n\\end{lm}\n\\begin{proof}\nLet $k$  be the positive integer s.t. $\\fg_k\\ne 0$ and $\\fg_{k+1}=\n0$. For $X,Y\\in\\fg$ denote by $X\\ast Y$ the Hausdorff product,\ni.e., $\\exp(X\\ast Y)=\\exp X\\exp Y$. It follows from the\nBaker-Campbell-Hausdorff formula that, for each  word $U$ in\n$\\{1,\\ldots,2m\\}$ of length at most $k$, there is $\\be_U\\in \\R$\ns.t.\n\\begin{equation}\\label{YmY2m}\n\\Bigl(\\sum_{i=1}^m Y_i\\Bigr)\\ast \\Bigl(\\sum_{i=m+1}^{2m}\nY_i\\Bigr)=\\sum_U \\be_U Y_U\n\\end{equation}\nfor every $Y_1,\\ldots,Y_{2m}\\in \\fg$.\n\nWrite $g_1=\\exp(\\sum_{i=1}^m t_ie_i)$ and $g_2=\\exp(\\sum_{i=1}^m\nt_{m+i}e_i)$. Substituting in \\eqref{YmY2m}, we obtain\n$$\n\\Bigl(\\sum_{i=1}^m t_ie_i\\Bigr)\\ast \\Bigl(\\sum_{i=1}^m\nt_{m+i}e_i\\Bigr)=\\sum_U \\be_U t_{u_1}\\cdots t_{u_s}E_U\\,,\n$$\nwhere $E_U=[e_{u_1},[e_{u_2},\\cdots, e_{u_s}]\\cdots]$. Denote\n$u_i$ modulo~$n$ by~$\\bar u_i$. Then $E_U\\in \\fg_{j(U)}$, where\n$j(U)\\!:=\\sum_{i=1}^s w_{\\bar u_i}$. So we have\n$$\nE_U=\\sum_{w_p\\ge j(U)} \\al_{U,p} e_p\n$$\nfor some $\\al_{U,p}$.\n\nNow write $g_1g_2=\\exp(\\sum_{i=1}^m t'_ie_i)$. Then the\ncoefficient $t'_p$ is bounded by\n$$\n\\sum_{U}|\\be_U|\\,|\\al_{U,p}|\\,|t_{u_1}\\cdots t_{u_s}|\\,.\n$$\nNote that $w_p\\ge j(U)$ implies\n$$\n|t_{u_1}\\cdots t_{u_s}|^{1\/w_p}\\le |t_{u_1}\\cdots\nt_{u_s}|^{1\/j(U)}+1\\le \\sum_{i=1}^s|t_{u_i}|^{1\/w_{\\bar\nu_i}}+1\\,.\n$$\nHence there exist $C'$ and $D'$ s.t.\n$$\n\\sum_{p=1}^m|t'_p|^{1\/w_p}\\le\nC'\\left(\\sum_{i=1}^m|t_i|^{1\/w_i}+\\sum_{i=1}^m|t_{m+i}|^{1\/w_i}\\right)+D'\\,,\n$$\nwhich implies \\eqref{sig12}.\n\\end{proof}\n\nFix subspaces $\\fv_1,\\ldots,\\fv_k$ s.t.\n$\\fv_i\\oplus\\fg_{i+1}=\\fg_i$. Evidently,\n$\\fg=\\oplus_{i=1}^k\\fv_i$.  It can easily be checked that this\ndecomposition can assumed compatible with an $\\mathscr{F}$-basis.\nSet $\\fv^{(1)}\\!:=\\fv_{1}$ and\n$\\fv^{(j)}\\!:=[\\fv_{1},\\fv^{(j-1)}]$ for $j>1$. It is not hard to\nsee that\n\\begin{equation}\\label{fgjk}\n\\fg_j=\\fv^{(j)}+\\cdots+ \\fv^{(k)}\\,.\n\\end{equation}\n\n\\begin{lm}\\label{normtrick}\nFor any $C>0$ there is a norm $\\|\\cdot\\|$ on $\\fg$ (as a real\nlinear space) s.t. the following conditions are satisfied.\n\\begin{enumerate}\n\\item If $V=\\sum_s V_s$, where $V_s\\in \\fv_s$, then $\\|V_s\\|\\le\n\\|V\\|$ for all $s$.\n\n\\item For any $p\\in\\N$, $u_1,\\ldots,u_p\\in \\{1,\\ldots,k\\}$, and $V_{u_s}\\in \\fv_{u_s}$\n$(s=1,\\ldots,p)$, one has\n$$\n\\|\\,[V_{u_1},[V_{u_2},\\cdots, V_{u_p}]\\cdots]\\,\\|\\le C\n\\|V_{u_1}\\|\\,\\|V_{u_2}\\|\\cdots \\|V_{u_p}\\|\\,.\n$$\n\\end{enumerate}\n\\end{lm}\n\\begin{proof}\nFix a norm $\\|\\cdot\\|_s$ on each $\\fv_s$ and consider norms on\n$\\fg$ of the form $\\|\\sum_s V_s\\|=\\sum \\la_s\\|V_s\\|_s$\n$(\\la_s>0)$. Obviously, any such norm satisfies to~(1).\nProceeding by induction on $p$ it is not hard to show that there\nis a norm of this form that satisfies to~(2).\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{eelUsi} (the real case)]\nLet $\\ell$ and $\\ell'$ be the word length functions corresponding\nto the generating sets\n$$\n\\bigcup_{i=1}^m\\{\\exp( t_ie_i)\\!:\\,|t_i|\\le 1\\}\\quad\\text{and}\\quad\\{\\exp(X)\\!:\\, X\\in\\fv_1,\\,\\|X\\|\\le 1\\}\\,,\n$$\nresp. The first set is generating, since for each $i$ the linear\nspan of $\\{e_i,\\ldots,e_m\\}$ is a subalgebra of $\\fg$. To see that\nthe second set is generating one can apply \\eqref{fgjk} with\n$j=1$, the surjectivity of the exponential map, and both parts of\nLemma~\\ref{fromV}. Since  $G$ is compactly generated, all word\nlength functions are equivalent \\cite[Th.~1.1.21]{Sch93} (cf. also\nCorollary~\\ref{eUdl} below). Thus, it suffices to show that\n$\\ell\\lesssim \\bar\\si \\lesssim \\si \\lesssim\\ell'$, where\n$\\lesssim$ means ''is dominated by''.\n\n(1) First, we prove that $\\ell\\lesssim \\bar\\si$. Denote by $S$\nthe subset of indices $p$ s.t. $w_p=1$ (eq., $e_p\\in\\fv_1$). For\nany word $U=(u_1,\\ldots,u_j)$ in $S$ set $\\la(U)=j$ and consider\nthe $j$th commutator $E_U(t)\\!:=[te_{u_1},[te_{u_2},\\cdots,\nte_{u_j}]\\cdots]$ for $t\\in\\R$.\n\nFix $j\\in\\{1,\\ldots,k\\}$ and $e_i$ with $w_i=j$. Since $e_i\\in\n\\fg_j$, it follows from \\eqref{fgjk} that $e_i$ is a linear\ncombination $\\sum_U\\mu_U E_U(1)$, where $U$ runs all words in $S$\nwith $j\\le \\la(U)\\le k$. Therefore, for any $t\\in\\R$,\n\\begin{equation}\\label{teinuU}\nte_i=\\sgn(t)\\, \\sum_{j\\le \\la(U)\\le k}\\mu_U\nE_U\\bigl(|t|^{1\/\\la(U)}\\bigr) \\,.\n\\end{equation}\nFor simplicity, we assume that $t>0$; for negative $t$ the\nfollowing estimates are the same.\n\nEnumerating all words in the sum above as $U_1,\\ldots,U_n$ and\napplying Part~(A) of Lemma~\\ref{fromV} for $n$, we can substitute\n$\\sum\\mu_U E_U(|t|^{1\/\\la(U)})$ in \\eqref{exppro} and get from\n\\eqref{teinuU} the equality\n\\begin{equation*}\n\\exp(te_i)=\\prod_{r=1}^N \\exp\\bigl(\\al_r\n\\mu_{U_r}E_{U_r}\\bigl(|t|^{1\/\\la(U_r)}\\bigr)\\bigr)\\,.\n\\end{equation*}\nFurther, applying Part~(B) of Lemma~\\ref{fromV} to each factor in\nthe product with $Y_1=\\al_r\\mu_{U_r} |t|^{1\/\\la(U_r)} e_1$ and\n$Y_s=|t|^{1\/\\la(U_r)} e_s$ for $s\\ge 2$, we have  that there are\n$N''$ and $\\be_1,\\ldots,\\be_{N''}$ independent in $t$ with some\npositive integers $\\la_1,\\ldots,\\la_{N''}$ non less that~$j$ s.t.\n\\begin{equation*}\n\\exp(te_i)=\\prod_{p=1}^{N''} \\exp\\bigl(|t|^{1\/\\la_p} \\be_p\ne_{i_p}\\bigr)\\,.\n\\end{equation*}\nHence, for each $t\\ge 0$,\n$$\n\\ell(\\exp te_i)\\le \\sum_p \\ell\\bigl(\\exp\\bigr(|t|^{1\/\\la_p} \\be_p\ne_{i_p}\\bigr)\\bigr)\\le \\sum_p (|t|^{1\/\\la_p} \\be_p+1) \\,.\n$$\nSince $|t|^{1\/\\la_p}\\le |t|^{1\/j}$ for $|t|\\ge 1$, we obtain\n$\\ell(\\exp(te_i))\\le C_j |t|^{1\/j}+D_j$ for all $t$, where $C_j$\nand $D_j$ depend only on $j$.\n\nFinally, write any $g\\in G$ as\n$$\ng=\\prod_{i=1}^m \\exp(\\bar t_i\\,e_i)\\,,\n$$\nwhere $\\bar{t}_1,\\ldots, \\bar{t}_ m\\in \\R$. Therefore\n$$\n\\ell(g)\\le \\sum_i \\ell(\\exp(\\bar t_i\\,e_i))\\le \\sum_i C_{w_i}\n|\\bar t_i|^{1\/{w_i}}+D_{w_i}\\,.\n$$\nThus, $\\ell$ is dominated by $\\bar\\si$.\n\n(2) Secondly, we prove that $\\bar\\si\\lesssim\\si$ (cf. the proof\nof \\cite[II.4.17]{DER}). We show by induction on $i$ in the\nreverse order that for each $i=1,\\ldots,m$ there are constants\n$A_i$ and $B_i$ s.t. for every $g=\\exp(\\sum_{s=i}^m t_se_s)$ the\nestimate $\\bar\\si(g)\\le A_i \\si(g)+B_i$ holds.\n\nObviously, if $i=m$, then $\\bar\\si(g)=\\si(g)$. Suppose that the\ninduction assumption is satisfied for $i$.  The\nBaker-Campbell-Hausdorff  formula implies that\n\\begin{equation}\\label{BCHfi1}\n (-t_{i-1}e_{i-1})\\ast\\Bigl(\\sum_{s=i-1}^m\nt_se_s\\Bigr)=\\sum_{s=i}^m t'_se_s\n\\end{equation}\nfor some $t'_s$. Write $\\exp(\\sum_{s=i}^m t'_se_s)=\\prod_{s=i}^m\n\\exp(\\bar t_s\\,e_s)$. Then\n$$\n\\exp\\Bigl(\\sum_{s=i-1}^m t_se_s\\Bigr)=\\prod_{s=i-1}^m \\exp(\\bar\nt_s\\,e_s)\\,,\n$$\nwhere $\\bar t_{i-1}\\!:=t_{i-1}$. Applying Lemma~\\ref{sisubpo} to\n\\eqref{BCHfi1}, we get\n$$\n\\max_{i\\le s\\le m}|t'_{s}|^{{1\/w_s}}\\le C\\max_{i-1\\le s\\le\nm}|t_{s}|^{{1\/w_s}} +D\\,.\n$$\nBy the inductive assumption, we have\n$$\n\\max_{i\\le s\\le m}|\\bar{t}_{s}|^{{1\/w_s}}\\le A_i\\max_{i\\le s\\le\nm}|t'_{s}|^{{1\/w_s}}+B_i\\,.\n$$\nCombining the two estimates we obtain\n$$\n\\max_{i-1\\le s\\le m}|\\bar{t}_{s}|^{{1\/w_s}}\\le\nA_{i-1}\\max_{i-1\\le s\\le m}|t_{s}|^{{1\/w_s}}+B_{i-1}\n$$\nfor some $A_{i-1}$ and $B_{i-1}$ depending only on $i$. The\ninduction is complete.\n\nFinally, note that we have shown that $\\bar\\si(g)\\le A_1\n\\si(g)+B_1$ for all $g\\in G$.\n\n\n(3) Thirdly, we prove that $\\si\\lesssim\\ell'$. Let $\\|\\cdot\\|$ be\nthe norm on $\\fg$ existing by Lemma~\\ref{normtrick} (the value of\nconstant $C$ is specified below). Note that $\\si(g)$ is equivalent\nto the function $g\\mapsto \\max_j\\|V_j\\|^{1\/j}$, where\n$g=\\exp(\\sum_j V_j$) with $V_j\\in\\fv_j$. So it suffices to show\nthat $\\ell'(g)=n$ implies $\\|V_j\\|\\le n^j$ for all~$j$.\n\nWe proceed by induction. For $n=0$ and $n=1$ the claim is\nobvious. Suppose that it holds for  $ n-1\\ge 1$. If $\\ell'(g)=n$,\nthen $g=g_1g_2$, where $\\ell'(g_1)=1$ and $\\ell'(g_2)= n-1$,\ni.e., $g_1=\\exp V_0$, where $V_0\\in\\fv_1$ with $\\|V_0\\|\\le 1$ and\n$g_2=\\exp (\\sum_j V_j)$, where $V_j\\in\\fv_j$ with $\\|V_j\\|\\le\n(n-1)^j$. Write $g_1g_2=\\exp(\\sum_{j=1}^k W_j)$, where\n$W_j\\in\\fv_j$. We need to show that $\\|W_j\\|\\le n^j$ for all $j$.\n\nNote that, by the Baker-Campbell-Hausdorff  formula, there are\n$\\ga_U$ s.t.\n\\begin{equation}\\label{corCBH}\nV_0\\ast(V_1+\\cdots+V_K)=\\exp\\Bigl(V_0+V_1+\\cdots+V_K+\\sum_U \\ga_U\nV_U\\Bigr)\\,,\n\\end{equation}\nfor  any $V_0\\in\\fv_1$ and $V_j\\in\\fv_j$ ($j=1,\\ldots,k$), where\n$V_U\\!:=[V_{u_1},[V_{u_2},\\cdots, V_{u_p}]$ and\n$U=(u_1,\\ldots,u_p)$ runs all words in $\\{0,1,\\ldots,k\\}$ of\nlength at least $2$ and at most $k$, and containing at least~$1$\nof occurrence of~$0$.\n\nFurther, for each $U$, there is a unique decomposition\n$$\nV_U=\\sum_{j=|U|}^k Y_{U,j}\\,;\\qquad(Y_{U,j}\\in\\fv_j)\\,,\n$$\nwhere $|U|$ denotes the sum of $u_1+\\cdots+u_p$ and the number of\noccurrence of~$0$ in~$U$. We get from \\eqref{corCBH} that\n$W_1=V_0+V_1$ and for $j\\ge 2$\n\\begin{equation}\\label{WjVkS}\nW_j=V_j+\\sum_{U\\in S_j} \\ga_U Y_{U,j}\\,,\n\\end{equation}\nwhere $S_j$ denotes the set of words $U$ s.t. the length of $U$ is\nat least $2$, $|U|\\le j$, and $U$ contains at least~$1$ of\noccurrence of~$0$.\n\nBy setting $C\\!:= (\\sum_{U} |\\ga_U|)^{-1}$  in\nLemma~\\ref{normtrick} we get\n\\begin{multline}\\label{gaYUje}\n\\Bigl\\|\\sum_{U\\in S_j} \\ga_U Y_{U,j}\\Bigr\\|\\le \\sum_{U\\in S_j} |\\ga_U|\\, \\|Y_{U,j}\\|\\le\\text{(by Part~(A))}\\\\\n \\sum_{U\\in S_j} |\\ga_U|\\,\n\\|V_{U}\\|\\le \\text{(by Part~(B))} \\\\\n\\sum_{U\\in S_j} |\\ga_U|\\, C \\,\\|V_{u_1}\\|\\,\\|V_{u_2}\\|\\cdots \\|V_{u_p}\\| \\le\\\\\n\\max_{U\\in S_j}\\{\\|V_{u_1}\\|\\,\\|V_{u_2}\\|\\cdots \\|V_{u_p}\\|\\}\\,.\n\\end{multline}\nAccording to the inductive assumption, $\\|V_{u_s}\\|\\le\n(n-1)^{u_s}$ for all $u_s$. Since $U\\in S_j$, we have\n$u_1+\\cdots+u_p< |U|\\le j$. Therefore,\n$$\\|V_{u_1}\\|\\,\\|V_{u_2}\\|\\cdots \\|V_{u_p}\\|\\le (n-1)^{j-1}\\,.$$\nFinally, \\eqref{WjVkS} and \\eqref{gaYUje} imply $\\|W_j\\|\\le\n2(n-1)^{j-1}\\le n^j$ for $j\\ge 2$ and obviously $\\|W_1\\|\\le\n\\|V_0\\|+\\|V_1\\|\\le n$.\n\\end{proof}\n\n\n\n\\subsection*{On left invariant Riemannian distances}\nThe following remarks explain why  Theorem~\\ref{eelUsi} can be\nreformulated in terms of left invariant (sub)-Riemannian\ndistances; thus, we have a direct connection with results\nin~\\cite{Be96,DER,Kar94,VSC92}.\n\nWe say that a  length function $\\ell$ is \\emph{symmetric} if\n$\\ell(e) = 0$ and $\\ell(g^{-1})=\\ell(g)$  for all $g\\in G$.\nObviously, for given  length function $\\ell$, the function $\\ell'$\ndefined by\n\\begin{equation*}\n\\ell'(g)\\!:=\\max\\{\\ell(g),\\,\\ell(g^{-1})\\},\\quad(g\\ne\ne)\\qquad\\text{and}\\quad \\ell'(e)\\!:=1\n\\end{equation*}\nis a symmetric length function.\n\nFor a symmetric length function $\\ell$ consider the following\ncondition:\n\n$(\\al)$ \\emph{there exists $C$ such that, for each $g\\in G$\nsatisfying $\\ell(g) \\ge 1$, there are $g_0,\\ldots,g_n\\in G$ s.t.\n$g_0 = e$, $g_n = g$, and $\\ell(g_i^{-1}g_{i+1}) \\le 1$ $(i =\n0,\\ldots, n-1)$ with $n \\le C\\ell(g)$.}\n\nIt is trivial that the formulas ${ }_l d(g,h)\\!:=\\ell(g^{-1}h)$\nand ${ }_d\\ell(g)\\!:=d(e,g)$ determine a bijection between the\nset of symmetric length functions on $G$  and the set of locally\nbounded left invariant distances on $G$. Under this\ncorrespondence our condition ($\\al$) becomes condition~$(C_3)$\nfrom p.~40 of  \\cite{VSC92}. In the terminology of\n\\cite[Def.~3.B.1]{CH16}, property~$(C_3)$ means that $d$ is\n\\emph{$1$-large-scale geodesic}.\n\n\\begin{pr}\\label{slfeq}\nLet $G$ be a locally compact group and let $\\ell_1$ and $\\ell_2$\nbe symmetric length functions on $G$ satisfying  to condition\n$(\\al)$. Suppose that $\\ell_1$ is bounded on $\\{g\\in G:\\,\n\\ell_2(g)\\le 1\\}$ and $\\ell_2$ is bounded on $\\{g\\in G:\\,\n\\ell_1(g)\\le 1\\}$. Then $\\ell_1$ and $\\ell_2$ are equivalent.\n\\end{pr}\n\\begin{proof}\nIt is noted in \\cite[Rem.~III.4.3]{VSC92} that the argument from\n[ibid., Prop.~III.4.2] can be applied in our situation.\n\\end{proof}\n\nRecall that every connected locally compact group is compactly\ngenerated \\cite[Th.~7.4]{HeRo}.\n\n\\begin{pr}\\label{eUdl}\nLet $G$ be a connected real Lie group. Suppose that $U$ is a\nsymmetric relatively compact generating set and $d$ is a distance\ndetermined by a left invariant Riemannian metric. Then $\\ell_U$\nis equivalent (at infinity) to~${\n}_d\\ell$.\n\\end{pr}\n\\begin{proof}\nIt is easy to see that $\\ell_U$ is bounded on $\\{g\\in G:\\, {\n}_d\\ell(g)\\le 1\\}$ and ${ }_d\\ell$ is bounded on $\\{g\\in\nG:\\,\\ell_U(g)\\le 1\\}$. By Proposition~\\ref{slfeq}, it suffices to\nshow that $\\ell_U$ and~${ }_d\\ell$ satisfy to $(\\al)$.\n\nLet $\\ell_U(g)=n>0$ and fix $g=h_1\\cdots h_n$ where $h_i\\in U$.\nSet $g_0=e$, and   $g_i=h_1\\cdots h_i$ for $i=1,\\ldots,n$. Then\n$\\ell_U(g_i^{-1}g_{i+1})=\\ell(h_{i+1}) \\le 1$ for $i<n$. Thus\n$(\\al)$ is satisfied for $\\ell_U$ with $C=1$. Since $d$ is\ndefined as the infimum of lengths of piecewise smooth paths,\ncondition $(\\al)$ is satisfied for ${ }_d\\ell$.\n\\end{proof}\n\nFor a nilpotent Lie group $G$, the distance  $d$ in\nProposition~\\ref{eUdl} can be replaced by a Carnot-Carath\\'eodory\ndistance in the sense\nof~\\cite[Sect.~III.4]{VSC92}.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}