12 Fourier analysis

12 Fourier analysis.ipynb

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      "source": "# Fourier解析\n\n黒木玄\n\n2018-06-28\n\n* Copyright 2018 Gen Kuroki\n* License: MIT https://opensource.org/licenses/MIT\n\nこのファイルは次の場所できれいに閲覧できる:\n\n* http://nbviewer.jupyter.org/github/genkuroki/Calculus/blob/master/12%20Fourier%20analysis.ipynb\n\nこのファイルは <a href=\"https://juliabox.com\">Julia Box</a> で利用できる.\n\n自分のパソコンに<a href=\"https://julialang.org/\">Julia言語</a>をインストールしたい場合には\n\n* <a href=\"http://nbviewer.jupyter.org/gist/genkuroki/81de23edcae631a995e19a2ecf946a4f\">WindowsへのJulia言語のインストール</a>\n\nを参照せよ.\n\n論理的に完璧な説明をするつもりはない. 細部のいい加減な部分は自分で訂正・修正せよ.\n\n$\n\\newcommand\\eps{\\varepsilon}\n\\newcommand\\ds{\\displaystyle}\n\\newcommand\\Z{{\\mathbb Z}}\n\\newcommand\\R{{\\mathbb R}}\n\\newcommand\\C{{\\mathbb C}}\n\\newcommand\\QED{\\text{□}}\n\\newcommand\\root{\\sqrt}\n\\newcommand\\bra{\\langle}\n\\newcommand\\ket{\\rangle}\n\\newcommand\\d{\\partial}\n\\newcommand\\sech{\\operatorname{sech}}\n\\newcommand\\sign{\\operatorname{sign}}\n$"
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      "source": "<h1>Table of Contents<span class=\"tocSkip\"></span></h1>\n<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#Riemann-Lebesgueの定理\" data-toc-modified-id=\"Riemann-Lebesgueの定理-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Riemann-Lebesgueの定理</a></span><ul class=\"toc-item\"><li><span><a href=\"#階段函数\" data-toc-modified-id=\"階段函数-1.1\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>階段函数</a></span></li><li><span><a href=\"#Riemann-Lebesgueの定理とその証明\" data-toc-modified-id=\"Riemann-Lebesgueの定理とその証明-1.2\"><span class=\"toc-item-num\">1.2&nbsp;&nbsp;</span>Riemann-Lebesgueの定理とその証明</a></span></li></ul></li><li><span><a href=\"#Fourier変換の逆変換の収束\" data-toc-modified-id=\"Fourier変換の逆変換の収束-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>Fourier変換の逆変換の収束</a></span><ul class=\"toc-item\"><li><span><a href=\"#Fourier変換とFourier逆変換の定義\" data-toc-modified-id=\"Fourier変換とFourier逆変換の定義-2.1\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span>Fourier変換とFourier逆変換の定義</a></span></li><li><span><a href=\"#Dirichlet核\" data-toc-modified-id=\"Dirichlet核-2.2\"><span class=\"toc-item-num\">2.2&nbsp;&nbsp;</span>Dirichlet核</a></span></li><li><span><a href=\"#Dirichlet積分の公式\" data-toc-modified-id=\"Dirichlet積分の公式-2.3\"><span class=\"toc-item-num\">2.3&nbsp;&nbsp;</span>Dirichlet積分の公式</a></span></li><li><span><a href=\"#Riemannの局所性定理\" data-toc-modified-id=\"Riemannの局所性定理-2.4\"><span class=\"toc-item-num\">2.4&nbsp;&nbsp;</span>Riemannの局所性定理</a></span></li><li><span><a href=\"#Fourier変換の逆変換の収束性-(Diniの条件)\" data-toc-modified-id=\"Fourier変換の逆変換の収束性-(Diniの条件)-2.5\"><span class=\"toc-item-num\">2.5&nbsp;&nbsp;</span>Fourier変換の逆変換の収束性 (Diniの条件)</a></span></li></ul></li><li><span><a href=\"#Fourier級数の収束\" data-toc-modified-id=\"Fourier級数の収束-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>Fourier級数の収束</a></span><ul class=\"toc-item\"><li><span><a href=\"#Fourier級数の定義\" data-toc-modified-id=\"Fourier級数の定義-3.1\"><span class=\"toc-item-num\">3.1&nbsp;&nbsp;</span>Fourier級数の定義</a></span></li><li><span><a href=\"#Fourier展開のDirichlet核\" data-toc-modified-id=\"Fourier展開のDirichlet核-3.2\"><span class=\"toc-item-num\">3.2&nbsp;&nbsp;</span>Fourier展開のDirichlet核</a></span></li><li><span><a href=\"#Fourier級数に関するRiemannの局所性定理\" data-toc-modified-id=\"Fourier級数に関するRiemannの局所性定理-3.3\"><span class=\"toc-item-num\">3.3&nbsp;&nbsp;</span>Fourier級数に関するRiemannの局所性定理</a></span></li><li><span><a href=\"#Fourier級数の収束-(Diniの条件)\" data-toc-modified-id=\"Fourier級数の収束-(Diniの条件)-3.4\"><span class=\"toc-item-num\">3.4&nbsp;&nbsp;</span>Fourier級数の収束 (Diniの条件)</a></span></li></ul></li><li><span><a href=\"#Poissonの和公式\" data-toc-modified-id=\"Poissonの和公式-4\"><span class=\"toc-item-num\">4&nbsp;&nbsp;</span>Poissonの和公式</a></span><ul class=\"toc-item\"><li><span><a href=\"#Poissonの和公式とその証明\" data-toc-modified-id=\"Poissonの和公式とその証明-4.1\"><span class=\"toc-item-num\">4.1&nbsp;&nbsp;</span>Poissonの和公式とその証明</a></span></li><li><span><a href=\"#モジュラー変換性を持つ函数の例\" data-toc-modified-id=\"モジュラー変換性を持つ函数の例-4.2\"><span class=\"toc-item-num\">4.2&nbsp;&nbsp;</span>モジュラー変換性を持つ函数の例</a></span></li></ul></li></ul></div>"
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      "source": "using Plots\ngr(); ENV[\"PLOTS_TEST\"] = \"true\"\n#clibrary(:colorcet)\nclibrary(:misc)\n\nfunction pngplot(P...; kwargs...)\n    sleep(0.1)\n    pngfile = tempname() * \".png\"\n    savefig(plot(P...; kwargs...), pngfile)\n    showimg(\"image/png\", pngfile)\nend\npngplot(; kwargs...) = pngplot(plot!(; kwargs...))\n\nshowimg(mime, fn) = open(fn) do f\n    base64 = base64encode(f)\n    display(\"text/html\", \"\"\"<img src=\"data:$mime;base64,$base64\">\"\"\")\nend\n\nusing SymPy\n#sympy[:init_printing](order=\"lex\") # default\n#sympy[:init_printing](order=\"rev-lex\")\n\nusing SpecialFunctions\nusing QuadGK",
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      "source": "## Riemann-Lebesgueの定理"
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      "source": "### 階段函数\n\n空でない区間 $I$ 上の函数で部分集合 $A$ 上で $1$, その外で $0$ になる函数を $\\chi_A(x)$ と書く:\n\n$$\n\\chi_A(x) = \\begin{cases}\n1 & (x\\in A) \\\\\n0 & (x\\not\\in A).\n\\end{cases}\n$$\n\n区間 $I$ に含まれる**長さが有限**の区間 $I_1,\\ldots,I_n$ と数 $\\alpha_1,\\ldots,\\alpha_n$ によって定まる函数\n\n$$\ns(x) = \\sum_{i=1}^n \\alpha_i\\chi_A(x) \\quad (x\\in I)\n$$\n\nを区間 $I$ 上の**階段函数**と呼ぶことにする.\n\n$f$ は区間 $I$ 上の函数であるとする. 任意の $\\eps>0$ に対してある階段函数 $s(x)$ で\n\n$$\n\\|f-s\\|_1 = \\int_I |f(x)-s(x)|\\,dx < \\eps\n$$\n\nを満たすものが存在するとき, $f$ は**階段函数で $L^1$ 近似可能**であるということにする. ここで $I=[a,b], [a,b), (a,b], (a,b)$ のとき $\\int_I=\\int_ab$ であるとする. $a=-\\infty$ であっても $b=\\infty$ であっても構わないとする."
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      "source": "**問題:** $\\R=(-\\infty,\\infty)$ 上の連続函数 $f(x)$ で $\\int_{-\\infty}^\\infty |f(x)|\\,dx < \\infty$ を満たすものは階段函数で $L^1$ 近似可能であることを示せ.\n\n**解答例:** 任意に $\\eps>0$ を取る $\\int_{-\\infty}^\\infty |f(x)|\\,dx < \\infty$ より, 十分に $R>0$ を大きくすると, \n\n$$\n\\int_{-\\infty}^{-R}|f(x)|\\,dx + \\int_R^\\infty |f(x)|\\,dx =\n\\int_{-\\infty}^\\infty |f(x)|\\,dx - \\int_{-R}^R |f(x)|\\,dx < \\frac{\\eps}{2}.\n$$\n\n$f(x)$ は $[-R,R]$ 上で一様連続なので, 正の整数 $N$ を十分に大きくして, 区間 $[-R,R]$ を $N$ 等分したものを $I_1,\\ldots,I_N$ として, $x_i\\in I_i$ を任意に取ると,  \n\n$$\n|f(x)-f(x_i)|< \\frac{\\eps}{4R} \\quad (x\\in I_i,\\ i=1,\\ldots,N).\n$$\n\nここで, 例えば $\\Delta = 2R/N$, $a_i=R+i\\Delta$, $I_1=[a_0,a_1)$, $\\ldots$, $I_{N-1}=[a_{N-2},a_{N-1}])$, $I_N=[a_{N-1},a_N]$ とおいたと考えてよい. このとき階段函数 $s(x)$ を\n\n$$\ns(x)=\\sum_{i=1}^N f(x_i)\\chi_{I_i}(x)\n$$\n\nと定めると,\n\n$$\n|f(x)-f(x_i)| < \\frac{\\eps}{4R} \\quad (-R\\leqq x\\leqq R).\n$$\n\nこのとき\n\n$$\n\\int_{-R}^R|f(x)-s(x)|\\,dx \\leqq \\frac{\\eps}{4R}2R = \\frac{\\eps}{2}.\n$$\n\nしたがって,\n\n$$\n\\begin{aligned}\n\\|f-s\\|_1 &= \\int_{-\\infty}^\\infty|f(x)-s(x)|\\,dx \n\\\\ &=\n\\int_{-\\infty}^{-R} |f(x)-s(x)|\\,dx + \\int_{-R}^R|f(x)-s(x)|\\,dx + \\int_R^\\infty|f(x)-s(x)|\\,dx \n\\\\ &=\n\\int_{-\\infty}^{-R} |f(x)|\\,dx + \\int_{-R}^R|f(x)-s(x)|\\,dx + \\int_R^\\infty|f(x)|\\,dx \n\\\\ &< \n\\frac{\\eps}{2}+\\frac{\\eps}{2}=\\eps.\n\\end{aligned}\n$$\n\nこれで示すべきことが示された. $\\QED$"
    },
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      "source": "**注意:** この注意はLebesgue積分論の知識を前提とする. 一般に区間上の可測函数 $f(x)$ で\n\n$$\n\\int_I |f(x)|\\,dx < \\infty\n$$\n\nを満たすものを, $I$ 上の **$L^1$ 函数** または**可積分函数**(もしくは**積分可能函数**)と呼ぶ. $I$ 上 $L_1$ 函数は $I$ 上の階段函数で $L^1$ 近似可能であることが知られている. $\\QED$"
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      "source": "**補題:** 区間 $I$ 上の階段函数 $s(x)$ について\n\n$$\n\\int_I s(x) e^{ipx}\\,dx \\to 0 \\quad (|p|\\to\\infty).\n$$\n\n**証明:** $-\\infty<a_j\\leqq b_j<\\infty$, $I_j=[a_j,b_j], [a_j,b_j), (a_j,b_j], (a_j,b_j)$ だとすると, $|p|\\to\\infty$ のとき,\n\n$$\n\\int_I \\chi_{I_j}(x)e^{ips}\\,dx = \n\\int_{a_j}^{b_j} e^{ipx}\\,dx = \n\\left[\\frac{e^{ipx}}{ip}\\right]_{x=a_j}^{x=b_j} =\n\\frac{e^{ipb_j} - e^{ipa_j}}{ip} \\to 0.\n$$\n\n階段函数は $\\chi_{I_j}(x)$ の有限一次結合なので補題の結果が成立することもわかる.  $\\QED$"
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      "source": "**問題** $a\\leqq b$ の取り方によらずに不等式\n\n$$\n\\left|\\int_a^b e^{ipx}\\,dx\\right|\\leqq \\frac{2}{|p|}\n$$\n\nが成立することを示し, このようなことが起こる理由について考えよ.\n\n**解答例:**\n$$\n\\left|\\int_a^b e^{ipx}\\,dx\\right| = \\left|\\frac{e^{ipb}-e^{ipa}}{ip}\\right| \\leqq \n\\frac{|e^{ipb}|+|e^{ipa}|}{|p|} = \\frac{2}{|p|}.\n$$\n\nただし, $\\leqq$ では三角不等式を使った.\n\n$e^{ipx}$ は周期 $2\\pi/|p|$ を持つ函数であり, ちょうど周期分だけ積分すると $0$ になる性質を持っている:\n\n$$\n\\int_c^{c+2\\pi/|p|} e^{ipx}\\,dx = 0.\n$$\n\nゆえに, 整数 $n$ で $a+2\\pi n/|p| \\leqq |b|$ となる最大のものを $N$ と書くと, \n\n$$\n\\int_a^b e^{ipx}\\,dx = \\int_{a+2\\pi n/|p|}^b e^{ipx}\\,dx, \\quad\nb - a+\\frac{2\\pi n}{|p|} < \\frac{2\\pi}{|p|}.\n$$\n\nこれより, \n\n$$\n\\left|\\int_a^b e^{ipx}\\,dx\\right| \\leqq \\int_{a+2\\pi n/|p|}^b \\,dx < \\frac{2\\pi}{|p|}.\n$$\n\nこれは上の問題の結果よりは荒い結果だが, この方法ではどのような仕組みで積分の絶対値の値が小さくなるかはよくわかる.  振動によってほとんど打ち消しあって消えて, 積分区間の端の部分で打ち消し合いが起こらない分だけ積分の値が生き残る. 打ち消し合いが不完全な積分区間の幅は小さくなり, 打ち消し合いが不完全な分だけ生き残った積分の値の絶対値も小さくなる. $\\QED$"
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      "source": "# e^{ipx} を sin(px) に置き換えた場合の図\n\na = 0.0\nb = 10.0\nf(p,x) = sin(p*x)\nmaxa(a,b,p) = a + 2π/abs(p)*fld(b-a, 2π/abs(p))\nx = a:0.01:b\nPP = []\nfor p in [5, 20]\n    xx = maxa(a,b,p):0.001:b\n    P = plot(title=\"p = $p\", titlefontsize=10)\n    plot!(legend=false, ylims=(-1.05,1.05))\n    plot!(x, f.(p,x))\n    hline!([0], color=:cyan, ls=:dot)\n    plot!(xx, f.(p,xx), color=:red, fill=(0, 0.5, :red))\n    push!(PP, P)\nend\nplot(PP..., size=(700, 250))",
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310.379,80.1777 310.408,80.5724 \n  310.437,80.9684 310.466,81.3656 310.495,81.7639 310.524,82.1634 310.553,82.5641 310.582,82.9659 310.611,83.3689 310.64,83.773 310.669,84.1782 310.698,84.5845 \n  310.728,84.992 310.757,85.4005 310.786,85.8101 310.815,86.2208 310.844,86.6326 310.873,87.0454 310.902,87.4592 310.931,87.8741 310.96,88.29 310.989,88.7069 \n  311.018,89.1248 311.047,89.5437 311.077,89.9635 311.106,90.3844 311.135,90.8062 311.164,91.2289 311.193,91.6526 311.222,92.0772 311.251,92.5027 311.28,92.9291 \n  311.309,93.3564 311.338,93.7846 311.367,94.2137 311.396,94.6437 311.425,95.0744 311.455,95.5061 311.484,95.9385 311.513,96.3718 311.542,96.8059 311.571,97.2407 \n  311.6,97.6764 311.629,98.1128 311.658,98.5501 311.687,98.988 311.716,99.4267 311.745,99.8661 311.774,100.306 311.804,100.747 311.833,101.189 311.862,101.631 \n  311.891,102.074 311.92,102.517 311.949,102.962 311.978,103.407 312.007,103.852 312.036,104.298 312.065,104.745 312.094,105.192 312.123,105.64 312.153,106.089 \n  312.182,106.538 312.211,106.987 312.24,107.438 312.269,107.888 312.298,108.34 312.327,108.791 312.356,109.244 312.385,109.696 312.414,110.149 312.443,110.603 \n  312.472,111.057 312.501,111.512 312.531,111.967 312.56,112.422 312.589,112.878 312.618,113.334 312.647,113.791 312.676,114.248 312.705,114.705 312.734,115.163 \n  312.763,115.621 312.792,116.079 312.821,116.537 312.85,116.996 312.88,117.456 312.909,117.915 312.938,118.375 312.967,118.835 312.996,119.295 313.025,119.756 \n  313.054,120.217 313.083,120.678 313.112,121.139 313.141,121.6 313.17,122.062 313.199,122.523 313.229,122.985 313.258,123.447 313.287,123.909 313.316,124.371 \n  313.345,124.834 313.374,125.296 313.403,125.759 313.432,126.221 313.461,126.684 313.49,127.147 313.519,127.609 313.548,128.072 313.577,128.535 313.607,128.998 \n  313.636,129.461 313.665,129.923 313.694,130.386 313.723,130.849 313.752,131.311 313.781,131.774 313.81,132.236 313.839,132.699 313.868,133.161 313.897,133.623 \n  313.926,134.085 313.956,134.547 313.985,135.009 314.014,135.471 314.043,135.932 314.072,136.393 314.101,136.855 314.13,137.315 314.159,137.776 314.188,138.237 \n  314.217,138.697 314.246,139.157 314.275,139.616 314.305,140.076 314.334,140.535 314.363,140.994 314.392,141.452 314.421,141.91 314.45,142.368 314.479,142.826 \n  314.508,143.283 314.537,143.74 314.566,144.196 314.595,144.652 314.624,145.108 314.654,145.563 314.683,146.018 314.712,146.472 314.741,146.926 314.77,147.38 \n  314.799,147.833 314.828,148.285 314.857,148.737 314.886,149.189 314.915,149.64 314.944,150.091 314.973,150.541 315.002,150.99 315.032,151.439 315.061,151.887 \n  315.09,152.335 315.119,152.782 315.148,153.229 315.177,153.674 315.206,154.12 315.235,154.564 315.264,155.008 315.293,155.452 315.322,155.894 315.351,156.336 \n  315.381,156.778 315.41,157.218 315.439,157.658 315.468,158.097 315.497,158.536 315.526,158.973 315.555,159.41 315.584,159.847 315.613,160.282 315.642,160.717 \n  315.671,161.15 315.7,161.583 315.73,162.015 315.759,162.447 315.788,162.877 315.817,163.307 315.846,163.736 315.875,164.163 315.904,164.59 315.933,165.017 \n  315.962,165.442 315.991,165.866 316.02,166.289 316.049,166.712 316.078,167.133 316.108,167.554 316.137,167.973 316.166,168.392 316.195,168.809 316.224,169.226 \n  316.253,169.641 316.282,170.056 316.311,170.469 316.34,170.882 316.369,171.293 316.398,171.703 316.427,172.113 316.457,172.521 316.486,172.928 316.515,173.334 \n  316.544,173.739 316.573,174.142 316.602,174.545 316.631,174.946 316.66,175.346 316.689,175.746 316.718,176.143 316.747,176.54 316.776,176.936 316.806,177.33 \n  316.835,177.723 316.864,178.115 316.893,178.506 316.922,178.895 316.951,179.283 316.98,179.67 317.009,180.056 317.038,180.44 317.067,180.823 317.096,181.205 \n  317.125,181.585 317.154,181.964 317.184,182.342 317.213,182.718 317.242,183.093 317.271,183.467 317.3,183.84 317.329,184.21 317.358,184.58 317.387,184.948 \n  317.416,185.315 317.445,185.68 317.474,186.044 317.503,186.407 317.533,186.768 317.562,187.127 317.591,187.485 317.62,187.842 317.649,188.197 317.678,188.551 \n  317.707,188.903 317.736,189.254 317.765,189.603 317.794,189.951 317.823,190.297 317.852,190.641 317.882,190.984 317.911,191.326 317.94,191.666 317.969,192.004 \n  317.998,192.341 318.027,192.676 318.056,193.01 318.085,193.342 318.114,193.672 318.143,194.001 318.172,194.328 318.201,194.653 318.231,194.977 318.26,195.299 \n  318.289,195.62 318.318,195.938 318.347,196.255 318.376,196.571 318.405,196.885 318.434,197.197 318.463,197.507 318.492,197.815 318.521,198.122 318.55,198.427 \n  318.579,198.731 318.609,199.032 318.638,199.332 318.667,199.63 318.696,199.927 318.725,200.221 318.754,200.514 318.783,200.805 318.812,201.094 318.841,201.381 \n  318.87,201.667 318.899,201.951 318.928,202.233 318.958,202.513 318.987,202.791 319.016,203.067 319.045,203.342 319.074,203.614 319.103,203.885 319.132,204.154 \n  319.161,204.421 319.19,204.686 319.219,204.949 319.248,205.21 319.277,205.47 319.307,205.727 319.336,205.983 319.365,206.236 319.394,206.488 319.423,206.738 \n  319.452,206.985 319.481,207.231 319.51,207.475 319.539,207.717 319.568,207.957 319.597,208.194 319.626,208.43 319.655,208.664 319.685,208.896 319.714,209.126 \n  319.743,209.354 319.772,209.58 319.801,209.804 319.83,210.026 319.859,210.245 319.888,210.463 319.917,210.679 319.946,210.893 319.975,211.104 320.004,211.314 \n  320.034,211.521 320.063,211.727 320.092,211.93 320.121,212.131 320.15,212.33 320.179,212.527 320.208,212.722 320.237,212.915 320.266,213.106 320.295,213.295 \n  320.324,213.481 320.353,213.666 320.383,213.848 320.412,214.028 320.441,214.206 320.47,214.382 320.499,214.556 320.528,214.727 320.557,214.897 320.586,215.064 \n  320.615,215.229 320.644,215.392 320.673,215.553 320.702,215.712 320.731,215.868 320.761,216.023 320.79,216.175 320.819,216.325 320.848,216.472 320.877,216.618 \n  320.906,216.761 320.935,216.902 320.964,217.041 320.993,217.178 321.022,217.312 321.051,217.445 321.08,217.575 321.11,217.703 321.139,217.828 321.168,217.952 \n  321.197,218.073 321.226,218.192 321.255,218.308 321.284,218.423 321.313,218.535 321.342,218.645 321.371,218.753 321.4,218.858 321.429,218.961 321.459,219.062 \n  321.488,219.161 321.517,219.257 321.546,219.352 321.575,219.443 321.604,219.533 321.633,219.62 321.662,219.706 321.691,219.788 321.72,219.869 321.749,219.947 \n  321.778,220.023 321.808,220.097 321.837,220.168 321.866,220.237 321.895,220.304 321.924,220.369 321.953,220.431 321.982,220.491 322.011,220.548 322.04,220.604 \n  322.069,220.657 322.098,220.708 322.127,220.756 322.156,220.802 322.186,220.846 322.215,220.888 322.244,220.927 322.273,220.964 322.302,220.998 322.331,221.031 \n  322.36,221.061 322.389,221.088 322.418,221.114 322.447,221.137 322.476,221.158 322.505,221.176 322.535,221.192 322.564,221.206 322.593,221.218 322.622,221.227 \n  322.651,221.234 322.68,221.238 322.709,221.24 322.738,221.24 322.767,221.238 322.796,221.233 322.825,221.226 322.854,221.217 322.884,221.205 322.913,221.192 \n  322.942,221.175 322.971,221.157 323,221.136 323.029,221.113 323.058,221.087 323.087,221.059 323.116,221.029 323.145,220.997 323.174,220.962 323.203,220.925 \n  323.232,220.886 323.262,220.844 323.291,220.8 323.32,220.754 323.349,220.705 323.378,220.655 323.407,220.601 323.436,220.546 323.465,220.488 323.494,220.428 \n  323.523,220.366 323.552,220.301 323.581,220.234 323.611,220.165 323.64,220.094 323.669,220.02 323.698,219.944 323.727,219.865 323.756,219.785 323.785,219.702 \n  323.814,219.617 323.843,219.529 323.872,219.439 323.901,219.347 323.93,219.253 323.96,219.157 323.989,219.058 324.018,218.957 324.047,218.854 324.076,218.748 \n  324.105,218.64 324.134,218.53 324.163,218.418 324.192,218.303 324.221,218.186 324.25,218.067 324.279,217.946 324.308,217.823 324.338,217.697 324.367,217.569 \n  324.396,217.439 324.425,217.306 324.454,217.172 324.483,217.035 324.512,216.896 324.541,216.755 324.57,216.611 324.599,216.466 324.628,216.318 324.657,216.168 \n  324.687,216.016 324.716,215.861 324.745,215.705 324.774,215.546 324.803,215.385 324.832,215.222 324.861,215.057 324.89,214.889 324.919,214.72 324.948,214.548 \n  324.977,214.374 325.006,214.198 325.036,214.02 325.065,213.84 325.094,213.657 325.123,213.473 325.152,213.286 325.181,213.098 325.21,212.907 325.239,212.714 \n  325.268,212.519 325.297,212.322 325.326,212.122 325.355,211.921 325.385,211.717 325.414,211.512 325.443,211.304 325.472,211.095 325.501,210.883 325.53,210.669 \n  325.559,210.454 325.588,210.236 325.617,210.016 325.646,209.794 325.675,209.57 325.704,209.344 325.733,209.116 325.763,208.886 325.792,208.654 325.821,208.42 \n  325.85,208.184 325.879,207.946 325.908,207.706 325.937,207.464 325.966,207.22 325.995,206.974 326.024,206.726 326.053,206.477 326.082,206.225 326.112,205.971 \n  326.141,205.716 326.17,205.458 326.199,205.199 326.228,204.937 326.257,204.674 326.286,204.409 326.315,204.142 326.344,203.873 326.373,203.602 326.402,203.329 \n  326.431,203.055 326.461,202.778 326.49,202.5 326.519,202.22 326.548,201.938 326.577,201.654 326.606,201.369 326.635,201.081 326.664,200.792 326.693,200.501 \n  326.722,200.208 326.751,199.914 326.78,199.617 326.809,199.319 326.839,199.019 326.868,198.717 326.897,198.414 326.926,198.109 326.955,197.802 326.984,197.493 \n  327.013,197.183 327.042,196.871 327.071,196.557 327.1,196.241 327.129,195.924 327.158,195.605 327.188,195.285 327.217,194.963 327.246,194.639 327.275,194.313 \n  327.304,193.986 327.333,193.657 327.362,193.327 327.391,192.995 327.42,192.661 327.449,192.326 327.478,191.989 327.507,191.651 327.537,191.311 327.566,190.969 \n  327.595,190.626 327.624,190.281 327.653,189.935 327.682,189.588 327.711,189.238 327.74,188.888 327.769,188.535 327.798,188.182 327.827,187.826 327.856,187.47 \n  327.885,187.111 327.915,186.752 327.944,186.391 327.973,186.028 328.002,185.664 328.031,185.299 328.06,184.932 328.089,184.564 328.118,184.194 328.147,183.823 \n  328.176,183.451 328.205,183.077 328.234,182.702 328.264,182.325 328.293,181.947 328.322,181.568 328.351,181.188 328.38,180.806 328.409,180.423 328.438,180.039 \n  328.467,179.653 328.496,179.266 328.525,178.878 328.554,178.488 328.583,178.098 328.613,177.706 328.642,177.313 328.671,176.918 328.7,176.523 328.729,176.126 \n  328.758,175.728 328.787,175.329 328.816,174.928 328.845,174.527 328.874,174.124 328.903,173.721 328.932,173.316 328.962,172.91 328.991,172.503 329.02,172.094 \n  329.049,171.685 329.078,171.275 329.107,170.863 329.136,170.451 329.165,170.037 329.194,169.623 329.223,169.207 329.252,168.791 329.281,168.373 329.31,167.955 \n  329.34,167.535 329.369,167.115 329.398,166.693 329.427,166.271 329.456,165.847 329.485,165.423 329.514,164.998 329.543,164.572 329.572,164.145 329.601,163.717 \n  329.63,163.288 329.659,162.858 329.689,162.428 329.718,161.996 329.747,161.564 329.776,161.131 329.805,160.697 329.834,160.263 329.863,159.827 329.892,159.391 \n  329.921,158.954 329.95,158.516 329.979,158.078 330.008,157.639 330.038,157.199 330.067,156.758 330.096,156.317 330.125,155.875 330.154,155.432 330.183,154.989 \n  330.212,154.545 330.241,154.1 330.27,153.655 330.299,153.209 330.299,128.682 330.27,128.682 330.241,128.682 330.212,128.682 330.183,128.682 330.154,128.682 \n  330.125,128.682 330.096,128.682 330.067,128.682 330.038,128.682 330.008,128.682 329.979,128.682 329.95,128.682 329.921,128.682 329.892,128.682 329.863,128.682 \n  329.834,128.682 329.805,128.682 329.776,128.682 329.747,128.682 329.718,128.682 329.689,128.682 329.659,128.682 329.63,128.682 329.601,128.682 329.572,128.682 \n  329.543,128.682 329.514,128.682 329.485,128.682 329.456,128.682 329.427,128.682 329.398,128.682 329.369,128.682 329.34,128.682 329.31,128.682 329.281,128.682 \n  329.252,128.682 329.223,128.682 329.194,128.682 329.165,128.682 329.136,128.682 329.107,128.682 329.078,128.682 329.049,128.682 329.02,128.682 328.991,128.682 \n  328.962,128.682 328.932,128.682 328.903,128.682 328.874,128.682 328.845,128.682 328.816,128.682 328.787,128.682 328.758,128.682 328.729,128.682 328.7,128.682 \n  328.671,128.682 328.642,128.682 328.613,128.682 328.583,128.682 328.554,128.682 328.525,128.682 328.496,128.682 328.467,128.682 328.438,128.682 328.409,128.682 \n  328.38,128.682 328.351,128.682 328.322,128.682 328.293,128.682 328.264,128.682 328.234,128.682 328.205,128.682 328.176,128.682 328.147,128.682 328.118,128.682 \n  328.089,128.682 328.06,128.682 328.031,128.682 328.002,128.682 327.973,128.682 327.944,128.682 327.915,128.682 327.885,128.682 327.856,128.682 327.827,128.682 \n  327.798,128.682 327.769,128.682 327.74,128.682 327.711,128.682 327.682,128.682 327.653,128.682 327.624,128.682 327.595,128.682 327.566,128.682 327.537,128.682 \n  327.507,128.682 327.478,128.682 327.449,128.682 327.42,128.682 327.391,128.682 327.362,128.682 327.333,128.682 327.304,128.682 327.275,128.682 327.246,128.682 \n  327.217,128.682 327.188,128.682 327.158,128.682 327.129,128.682 327.1,128.682 327.071,128.682 327.042,128.682 327.013,128.682 326.984,128.682 326.955,128.682 \n  326.926,128.682 326.897,128.682 326.868,128.682 326.839,128.682 326.809,128.682 326.78,128.682 326.751,128.682 326.722,128.682 326.693,128.682 326.664,128.682 \n  326.635,128.682 326.606,128.682 326.577,128.682 326.548,128.682 326.519,128.682 326.49,128.682 326.461,128.682 326.431,128.682 326.402,128.682 326.373,128.682 \n  326.344,128.682 326.315,128.682 326.286,128.682 326.257,128.682 326.228,128.682 326.199,128.682 326.17,128.682 326.141,128.682 326.112,128.682 326.082,128.682 \n  326.053,128.682 326.024,128.682 325.995,128.682 325.966,128.682 325.937,128.682 325.908,128.682 325.879,128.682 325.85,128.682 325.821,128.682 325.792,128.682 \n  325.763,128.682 325.733,128.682 325.704,128.682 325.675,128.682 325.646,128.682 325.617,128.682 325.588,128.682 325.559,128.682 325.53,128.682 325.501,128.682 \n  325.472,128.682 325.443,128.682 325.414,128.682 325.385,128.682 325.355,128.682 325.326,128.682 325.297,128.682 325.268,128.682 325.239,128.682 325.21,128.682 \n  325.181,128.682 325.152,128.682 325.123,128.682 325.094,128.682 325.065,128.682 325.036,128.682 325.006,128.682 324.977,128.682 324.948,128.682 324.919,128.682 \n  324.89,128.682 324.861,128.682 324.832,128.682 324.803,128.682 324.774,128.682 324.745,128.682 324.716,128.682 324.687,128.682 324.657,128.682 324.628,128.682 \n  324.599,128.682 324.57,128.682 324.541,128.682 324.512,128.682 324.483,128.682 324.454,128.682 324.425,128.682 324.396,128.682 324.367,128.682 324.338,128.682 \n  324.308,128.682 324.279,128.682 324.25,128.682 324.221,128.682 324.192,128.682 324.163,128.682 324.134,128.682 324.105,128.682 324.076,128.682 324.047,128.682 \n  324.018,128.682 323.989,128.682 323.96,128.682 323.93,128.682 323.901,128.682 323.872,128.682 323.843,128.682 323.814,128.682 323.785,128.682 323.756,128.682 \n  323.727,128.682 323.698,128.682 323.669,128.682 323.64,128.682 323.611,128.682 323.581,128.682 323.552,128.682 323.523,128.682 323.494,128.682 323.465,128.682 \n  323.436,128.682 323.407,128.682 323.378,128.682 323.349,128.682 323.32,128.682 323.291,128.682 323.262,128.682 323.232,128.682 323.203,128.682 323.174,128.682 \n  323.145,128.682 323.116,128.682 323.087,128.682 323.058,128.682 323.029,128.682 323,128.682 322.971,128.682 322.942,128.682 322.913,128.682 322.884,128.682 \n  322.854,128.682 322.825,128.682 322.796,128.682 322.767,128.682 322.738,128.682 322.709,128.682 322.68,128.682 322.651,128.682 322.622,128.682 322.593,128.682 \n  322.564,128.682 322.535,128.682 322.505,128.682 322.476,128.682 322.447,128.682 322.418,128.682 322.389,128.682 322.36,128.682 322.331,128.682 322.302,128.682 \n  322.273,128.682 322.244,128.682 322.215,128.682 322.186,128.682 322.156,128.682 322.127,128.682 322.098,128.682 322.069,128.682 322.04,128.682 322.011,128.682 \n  321.982,128.682 321.953,128.682 321.924,128.682 321.895,128.682 321.866,128.682 321.837,128.682 321.808,128.682 321.778,128.682 321.749,128.682 321.72,128.682 \n  321.691,128.682 321.662,128.682 321.633,128.682 321.604,128.682 321.575,128.682 321.546,128.682 321.517,128.682 321.488,128.682 321.459,128.682 321.429,128.682 \n  321.4,128.682 321.371,128.682 321.342,128.682 321.313,128.682 321.284,128.682 321.255,128.682 321.226,128.682 321.197,128.682 321.168,128.682 321.139,128.682 \n  321.11,128.682 321.08,128.682 321.051,128.682 321.022,128.682 320.993,128.682 320.964,128.682 320.935,128.682 320.906,128.682 320.877,128.682 320.848,128.682 \n  320.819,128.682 320.79,128.682 320.761,128.682 320.731,128.682 320.702,128.682 320.673,128.682 320.644,128.682 320.615,128.682 320.586,128.682 320.557,128.682 \n  320.528,128.682 320.499,128.682 320.47,128.682 320.441,128.682 320.412,128.682 320.383,128.682 320.353,128.682 320.324,128.682 320.295,128.682 320.266,128.682 \n  320.237,128.682 320.208,128.682 320.179,128.682 320.15,128.682 320.121,128.682 320.092,128.682 320.063,128.682 320.034,128.682 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          },
          "execution_count": 2,
          "metadata": {},
          "output_type": "execute_result"
        }
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    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### Riemann-Lebesgueの定理とその証明\n\n**定理(Riemann-Lebesgue):** $f$ は区間 $I$ 上の階段函数で $L^1$ 近似可能な函数であるとする. (Lebesgue積分論を知っている人は, $f(x)$ は区間 $I$ 上の $L^1$ 函数であると仮定してもよい.) このとき,\n\n$$\n\\int_I f(x)e^{ipx}\\,dx \\to 0 \\quad (|p|\\to\\infty).\n$$\n\n**証明:** 任意に $\\eps>0$ を取る. $f(x)$ は階段函数で $L^1$ 近似可能なので, ある階段函数 $\\ds s(x)=\\sum_{i=1}^n a_i\\chi_{I_i}(x)$ で\n\n$$\n\\|f-s\\|_1=\\int_I |f(x)-s(x)|\\,dx < \\frac{\\eps}{2}\n$$\n\nを満たすものが存在する. 階段函数に関する補題より, ある $R>0$ が存在して, $|p|\\geqq R$ ならば\n\n$$\n\\left|\\int_I s(x)e^{ipx}\\,dx\\right|<\\frac{\\eps}{2}.\n$$\n\nそのとき, \n\n$$\n\\begin{aligned}\n\\left|\\int_I f(x)e^{ipx}\\,dx\\right| &=\n\\left|\\int_I (f(x)-s(x))e^{ipx}\\,dx  + \\int_I s(x)e^{ipx}\\,dx\\right| \n\\\\ &\\leqq\n\\left|\\int_I (f(x)-s(x))e^{ipx}\\,dx\\right| + \\left|\\int_I s(x)e^{ipx}\\,dx\\right| \n\\\\ &\\leqq\n\\int_I |f(x)-s(x)|\\,dx + \\left|\\int_I s(x)e^{ipx}\\,dx\\right| \n\\\\ &=\n\\|f-s\\|_1 + \\left|\\int_I s(x)e^{ipx}\\,dx\\right| <\n\\frac{\\eps}{2}+\\frac{\\eps}{2} = \\eps.\n\\end{aligned}\n$$\n\nこれで $|p|\\to\\infty$ で $\\int_I f(x)e^{ipx}\\,dx\\to 0$ となることがわかった. $\\QED$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "**解説:** Riemann-Lebesgueの定理(リーマン・ルベーグの定理)は $|p|$ を大きくすると, $e^{ipx}$ が $x$ についてより細かく振動するようになるので, $e^{ipx}$ と積を取ってから積分すると, 実部と虚部それぞれの正の成分と負の成分が互いに打ち消され易くなって, 積分の絶対値の値が小さくなることを意味している. $\\QED$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "## Fourier変換の逆変換の収束"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### Fourier変換とFourier逆変換の定義\n\n$\\R$ 上の函数 $f(x)$ の**Fourier変換** $\\hat{f}(p)$ を次のように定義する:\n\n$$\n\\hat{f}(p) = \\int_\\R f(x)e^{-2\\pi ipx}\\,dx.\n$$\n\nさらに $\\R$ 上の函数 $g(p)$ の**Fourier逆変換** $\\check{g}(x)$ を次のように定義する:\n\n$$\n\\check{g}(x) = \\int_\\R g(p)e^{2\\pi ipx}\\,dp.\n$$\n\n$f(x)$ が $\\R$ 上の $L^1$ 函数ならば, Lebesgueの収束定理より $\\hat{f}(p)$ は $p$ の連続函数になり, Riemann-Lebegueの定理より $|p|\\to\\infty$ で $\\hat{f}(p)\\to 0$ となる. \n\n**注意:** Fourier変換とFourier逆変換を, 上の公式における $p$ を $p/(2\\pi)$ で置き換えることによって, \n\n$$\n\\hat{f}(p) = \\int_\\R f(x)e^{-ipx}\\,dx, \\quad\n\\check{g}(p) = \\frac{1}{2\\pi}\\int_\\R g(p)e^{ipx}\\,dp\n$$\n\nと定義するスタイルもよく使われている. $\\QED$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "この節では, $L^1$ 函数 $f(x)$ に対して, \n\n$$\nf(x) = \\lim_{N\\to\\infty} \\int_{-N}^N \\hat{f}(p)e^{ipx}\\,dp \n$$\n\nがいつ成立しているかを調べる. "
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### Dirichlet核\n\n**Dirichlet核** $D_N(x)$ を\n\n$$\nD_N(x) = \\int_{-N}^N e^{2\\pi ipx}\\,dp = \\left[\\frac{e^{2\\pi ipx}}{2\\pi ix}\\right]_{p=-N}^{p=N} =\n\\frac{e^{e\\pi iNx}-e^{-2\\pi iNx}}{2\\pi ix} = \\frac{\\sin(2\\pi Nx)}{\\pi x}\n$$\n\nと定める. ただし, $D_N(0)=2N$ と定めておく. $D_N(x)$ は有界な偶函数になる.\n\nこのとき\n\n$$\n\\begin{aligned}\n\\int_{-N}^N \\hat{f}(p)e^{2\\pi ipx}\\,dp &=\n\\int_{-N}^N \\left(\\int_\\R f(y)e^{-2\\pi ipy}\\,dy\\right)e^{2\\pi ipx}\\,dp =\n\\int_\\R f(y) \\left(\\int_{-N}^N e^{2\\pi ip(x-y)}\\,dp\\right)\\,dy\n\\\\ &=\n\\int_\\R f(y) D_N(x-y)\\,dy =\n\\int_\\R f(x+y) D_N(y)\\,dy.\n\\end{aligned}\n$$\n\n最後の等号で $y$ を $x+y$ で置き換えて, $D_N(-y)=D_N(y)$ を使った."
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "**問題:** Dirichlet核のグラフを描け. $\\QED$\n\n**解説:** $N$ が大きくなると, Dirichelt核 $D_N(x)$ のグラフの振動の仕方は細かくなり, $x=0$ の近くでの値も大きくなる.\n\n次のセルを見よ. $\\QED$"
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "# ディリクレ核 D_N(x) のグラフ\n\nD(N,x) = iszero(x) ? 2N : sin(2π*N*x)/(π*x)\nPP = []\nfor N in [1,2,3,4,5,6]\n    x = -6:0.01:6\n    P = plot(x, D.(N,x), title=\"N=$N\", titlefontsize=10)\n    push!(PP, P)\nend\nplot(PP..., size=(750, 400), legend=false)",
      "execution_count": 3,
      "outputs": [
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306.174,352.609 306.335,352.566 306.496,352.605 306.657,352.723 306.818,352.907 306.979,353.141 \n  307.14,353.401 307.301,353.662 307.462,353.899 307.623,354.088 307.785,354.21 307.946,354.254 308.107,354.214 308.268,354.094 308.429,353.905 308.59,353.667 \n  308.751,353.401 308.912,353.134 309.073,352.892 309.234,352.698 309.395,352.573 309.556,352.529 309.717,352.57 309.878,352.692 310.039,352.885 310.2,353.129 \n  310.361,353.401 310.522,353.674 310.684,353.921 310.845,354.119 311.006,354.247 311.167,354.293 311.328,354.251 311.489,354.126 311.65,353.929 311.811,353.679 \n  311.972,353.401 312.133,353.121 312.294,352.868 312.455,352.666 312.616,352.534 312.777,352.488 312.938,352.53 313.099,352.659 313.26,352.86 313.421,353.116 \n  313.583,353.401 313.744,353.687 313.905,353.946 314.066,354.154 314.227,354.288 314.388,354.336 314.549,354.292 314.71,354.161 314.871,353.954 315.032,353.692 \n  315.193,353.401 315.354,353.108 315.515,352.842 315.676,352.629 315.837,352.492 315.998,352.443 316.159,352.487 316.321,352.622 316.482,352.833 316.643,353.102 \n  316.804,353.401 316.965,353.701 317.126,353.974 317.287,354.191 317.448,354.333 317.609,354.383 317.77,354.337 317.931,354.2 318.092,353.983 318.253,353.707 \n  318.414,353.401 318.575,353.092 318.736,352.813 318.897,352.59 319.058,352.445 319.22,352.393 319.381,352.44 319.542,352.581 319.703,352.804 319.864,353.086 \n  320.025,353.401 320.186,353.717 320.347,354.004 320.508,354.233 320.669,354.382 320.83,354.435 320.991,354.387 321.152,354.242 321.313,354.014 321.474,353.724 \n  321.635,353.401 321.796,353.076 321.957,352.781 322.119,352.545 322.28,352.392 322.441,352.338 322.602,352.387 322.763,352.536 322.924,352.771 323.085,353.069 \n  323.246,353.401 323.407,353.735 323.568,354.038 323.729,354.28 323.89,354.437 324.051,354.494 324.212,354.443 324.373,354.29 324.534,354.049 324.695,353.742 \n  324.856,353.401 325.018,353.057 325.179,352.745 325.34,352.496 325.501,352.334 325.662,352.276 325.823,352.328 325.984,352.485 326.145,352.734 326.306,353.049 \n  326.467,353.401 326.628,353.754 326.789,354.076 326.95,354.332 327.111,354.499 327.272,354.559 327.433,354.506 327.594,354.343 327.756,354.088 327.917,353.763 \n  328.078,353.401 328.239,353.036 328.4,352.705 328.561,352.441 328.722,352.269 328.883,352.207 329.044,352.262 329.205,352.429 329.366,352.692 329.527,353.027 \n  329.688,353.401 329.849,353.777 330.01,354.118 330.171,354.391 330.332,354.569 330.493,354.633 330.655,354.576 330.816,354.404 330.977,354.132 331.138,353.786 \n  331.299,353.401 331.46,353.013 331.621,352.66 331.782,352.378 331.943,352.195 332.104,352.128 332.265,352.187 332.426,352.365 332.587,352.645 332.748,353.002 \n  332.909,353.401 333.07,353.802 333.231,354.166 333.392,354.458 333.554,354.647 333.715,354.716 333.876,354.656 334.037,354.472 334.198,354.182 334.359,353.813 \n  334.52,353.401 334.681,352.986 334.842,352.609 335.003,352.307 335.164,352.11 335.325,352.039 335.486,352.101 335.647,352.291 335.808,352.592 335.969,352.974 \n  336.13,353.401 336.292,353.831 336.453,354.221 336.614,354.534 336.775,354.738 336.936,354.812 337.097,354.748 337.258,354.551 337.419,354.239 337.58,353.843 \n  337.741,353.401 337.902,352.955 338.063,352.55 338.224,352.225 338.385,352.013 338.546,351.936 338.707,352.003 338.868,352.207 339.029,352.53 339.191,352.941 \n  339.352,353.401 339.513,353.864 339.674,354.285 339.835,354.622 339.996,354.842 340.157,354.922 340.318,354.854 340.479,354.642 340.64,354.306 340.801,353.878 \n  340.962,353.401 341.123,352.919 341.284,352.481 341.445,352.13 341.606,351.901 341.767,351.817 341.928,351.888 342.09,352.109 342.251,352.458 342.412,352.903 \n  342.573,353.401 342.734,353.902 342.895,354.359 343.056,354.725 343.217,354.964 343.378,355.052 343.539,354.978 343.7,354.748 343.861,354.384 344.022,353.92 \n  344.183,353.401 344.344,352.877 344.505,352.4 344.666,352.018 344.827,351.768 344.989,351.676 345.15,351.753 345.311,351.993 345.472,352.373 345.633,352.858 \n  345.794,353.401 345.955,353.948 346.116,354.447 346.277,354.847 346.438,355.109 346.599,355.206 346.76,355.125 346.921,354.875 347.082,354.477 347.243,353.969 \n  347.404,353.401 347.565,352.827 347.727,352.304 347.888,351.884 348.049,351.609 348.21,351.508 348.371,351.592 348.532,351.854 348.693,352.272 348.854,352.804 \n  349.015,353.401 349.176,354.003 349.337,354.553 349.498,354.994 349.659,355.284 349.82,355.391 349.981,355.303 350.142,355.027 350.303,354.589 350.464,354.029 \n  350.626,353.401 350.787,352.766 350.948,352.188 351.109,351.722 351.27,351.417 351.431,351.303 351.592,351.395 351.753,351.685 351.914,352.148 352.075,352.738 \n  352.236,353.401 352.397,354.071 352.558,354.682 352.719,355.174 352.88,355.498 353.041,355.618 353.202,355.522 353.363,355.215 353.525,354.727 353.686,354.102 \n  353.847,353.401 354.008,352.691 354.169,352.043 354.33,351.521 354.491,351.178 354.652,351.049 354.813,351.15 354.974,351.475 355.135,351.993 355.296,352.656 \n  355.457,353.401 355.618,354.155 355.779,354.844 355.94,355.4 356.101,355.766 356.263,355.904 356.424,355.797 356.585,355.453 356.746,354.901 356.907,354.195 \n  357.068,353.401 357.229,352.596 357.39,351.86 357.551,351.265 357.712,350.873 357.873,350.725 358.034,350.838 358.195,351.205 358.356,351.795 358.517,352.55 \n  358.678,353.401 358.839,354.263 359,355.053 359.162,355.692 359.323,356.114 359.484,356.275 359.645,356.155 359.806,355.761 359.967,355.129 360.128,354.316 \n  360.289,353.401 360.45,352.471 360.611,351.619 360.772,350.929 360.933,350.472 361.094,350.296 361.255,350.425 361.416,350.848 361.577,351.531 361.738,352.41 \n  361.899,353.401 362.061,354.408 362.222,355.334 362.383,356.084 362.544,356.582 362.705,356.775 362.866,356.638 363.027,356.179 363.188,355.437 363.349,354.481 \n  363.51,353.401 363.671,352.301 363.832,351.289 363.993,350.467 364.154,349.919 364.315,349.705 364.476,349.852 364.637,350.353 364.798,351.165 364.96,352.214 \n  365.121,353.401 365.282,354.612 365.443,355.728 365.604,356.637 365.765,357.245 365.926,357.485 366.087,357.327 366.248,356.776 366.409,355.88 366.57,354.718 \n  366.731,353.401 366.892,352.053 367.053,350.809 367.214,349.792 367.375,349.11 367.536,348.836 367.698,349.007 367.859,349.618 368.02,350.619 368.181,351.92 \n  368.342,353.401 368.503,354.919 368.664,356.325 368.825,357.478 368.986,358.257 369.147,358.575 369.308,358.388 369.469,357.701 369.63,356.569 369.791,355.09 \n  369.952,353.401 370.113,351.663 370.274,350.047 370.435,348.715 370.597,347.809 370.758,347.431 370.919,347.634 371.08,348.418 371.241,349.722 371.402,351.435 \n  371.563,353.401 371.724,355.433 371.885,357.333 372.046,358.908 372.207,359.991 372.368,360.456 372.529,360.235 372.69,359.324 372.851,357.787 373.012,355.752 \n  373.173,353.401 373.334,350.954 373.496,348.649 373.657,346.721 373.818,345.378 373.979,344.778 374.14,345.013 374.301,346.1 374.462,347.97 374.623,350.476 \n  374.784,353.401 374.945,356.475 375.106,359.403 375.267,361.885 375.428,363.652 375.589,364.487 375.75,364.255 375.911,362.914 376.072,360.528 376.234,357.269 \n  376.395,353.401 376.556,349.266 376.717,345.255 376.878,341.774 377.039,339.207 377.2,337.879 377.361,338.024 377.522,339.752 377.683,343.033 377.844,347.691 \n  378.005,353.401 378.166,359.712 378.327,366.072 378.488,371.867 378.649,376.466 378.81,379.27 378.971,379.761 379.133,377.549 379.294,372.407 379.455,364.302 \n  379.616,353.401 379.777,340.077 379.938,324.891 380.099,308.554 380.26,291.894 380.421,275.794 380.582,261.14 380.743,248.759 380.904,239.36 381.065,233.491 \n  381.226,231.496 381.387,233.491 381.548,239.36 381.709,248.759 381.87,261.14 382.032,275.794 382.193,291.894 382.354,308.554 382.515,324.891 382.676,340.077 \n  382.837,353.401 382.998,364.302 383.159,372.407 383.32,377.549 383.481,379.761 383.642,379.27 383.803,376.466 383.964,371.867 384.125,366.072 384.286,359.712 \n  384.447,353.401 384.608,347.691 384.77,343.033 384.931,339.752 385.092,338.024 385.253,337.879 385.414,339.207 385.575,341.774 385.736,345.255 385.897,349.266 \n  386.058,353.401 386.219,357.269 386.38,360.528 386.541,362.914 386.702,364.255 386.863,364.487 387.024,363.652 387.185,361.885 387.346,359.403 387.507,356.475 \n  387.669,353.401 387.83,350.476 387.991,347.97 388.152,346.1 388.313,345.013 388.474,344.778 388.635,345.378 388.796,346.721 388.957,348.649 389.118,350.954 \n  389.279,353.401 389.44,355.752 389.601,357.787 389.762,359.324 389.923,360.235 390.084,360.456 390.245,359.991 390.406,358.908 390.568,357.333 390.729,355.433 \n  390.89,353.401 391.051,351.435 391.212,349.722 391.373,348.418 391.534,347.634 391.695,347.431 391.856,347.809 392.017,348.715 392.178,350.047 392.339,351.663 \n  392.5,353.401 392.661,355.09 392.822,356.569 392.983,357.701 393.144,358.388 393.305,358.575 393.467,358.257 393.628,357.478 393.789,356.325 393.95,354.919 \n  394.111,353.401 394.272,351.92 394.433,350.619 394.594,349.618 394.755,349.007 394.916,348.836 395.077,349.11 395.238,349.792 395.399,350.809 395.56,352.053 \n  395.721,353.401 395.882,354.718 396.043,355.88 396.205,356.776 396.366,357.327 396.527,357.485 396.688,357.245 396.849,356.637 397.01,355.728 397.171,354.612 \n  397.332,353.401 397.493,352.214 397.654,351.165 397.815,350.353 397.976,349.852 398.137,349.705 398.298,349.919 398.459,350.467 398.62,351.289 398.781,352.301 \n  398.942,353.401 399.104,354.481 399.265,355.437 399.426,356.179 399.587,356.638 399.748,356.775 399.909,356.582 400.07,356.084 400.231,355.334 400.392,354.408 \n  400.553,353.401 400.714,352.41 400.875,351.531 401.036,350.848 401.197,350.425 401.358,350.296 401.519,350.472 401.68,350.929 401.841,351.619 402.003,352.471 \n  402.164,353.401 402.325,354.316 402.486,355.129 402.647,355.761 402.808,356.155 402.969,356.275 403.13,356.114 403.291,355.692 403.452,355.053 403.613,354.263 \n  403.774,353.401 403.935,352.55 404.096,351.795 404.257,351.205 404.418,350.838 404.579,350.725 404.741,350.873 404.902,351.265 405.063,351.86 405.224,352.596 \n  405.385,353.401 405.546,354.195 405.707,354.901 405.868,355.453 406.029,355.797 406.19,355.904 406.351,355.766 406.512,355.4 406.673,354.844 406.834,354.155 \n  406.995,353.401 407.156,352.656 407.317,351.993 407.478,351.475 407.64,351.15 407.801,351.049 407.962,351.178 408.123,351.521 408.284,352.043 408.445,352.691 \n  408.606,353.401 408.767,354.102 408.928,354.727 409.089,355.215 409.25,355.522 409.411,355.618 409.572,355.498 409.733,355.174 409.894,354.682 410.055,354.071 \n  410.216,353.401 410.377,352.738 410.539,352.148 410.7,351.685 410.861,351.395 411.022,351.303 411.183,351.417 411.344,351.722 411.505,352.188 411.666,352.766 \n  411.827,353.401 411.988,354.029 412.149,354.589 412.31,355.027 412.471,355.303 412.632,355.391 412.793,355.284 412.954,354.994 413.115,354.553 413.276,354.003 \n  413.438,353.401 413.599,352.804 413.76,352.272 413.921,351.854 414.082,351.592 414.243,351.508 414.404,351.609 414.565,351.884 414.726,352.304 414.887,352.827 \n  415.048,353.401 415.209,353.969 415.37,354.477 415.531,354.875 415.692,355.125 415.853,355.206 416.014,355.109 416.176,354.847 416.337,354.447 416.498,353.948 \n  416.659,353.401 416.82,352.858 416.981,352.373 417.142,351.993 417.303,351.753 417.464,351.676 417.625,351.768 417.786,352.018 417.947,352.4 418.108,352.877 \n  418.269,353.401 418.43,353.92 418.591,354.384 418.752,354.748 418.913,354.978 419.075,355.052 419.236,354.964 419.397,354.725 419.558,354.359 419.719,353.902 \n  419.88,353.401 420.041,352.903 420.202,352.458 420.363,352.109 420.524,351.888 420.685,351.817 420.846,351.901 421.007,352.13 421.168,352.481 421.329,352.919 \n  421.49,353.401 421.651,353.878 421.812,354.306 421.974,354.642 422.135,354.854 422.296,354.922 422.457,354.842 422.618,354.622 422.779,354.285 422.94,353.864 \n  423.101,353.401 423.262,352.941 423.423,352.53 423.584,352.207 423.745,352.003 423.906,351.936 424.067,352.013 424.228,352.225 424.389,352.55 424.55,352.955 \n  424.712,353.401 424.873,353.843 425.034,354.239 425.195,354.551 425.356,354.748 425.517,354.812 425.678,354.738 425.839,354.534 426,354.221 426.161,353.831 \n  426.322,353.401 426.483,352.974 426.644,352.592 426.805,352.291 426.966,352.101 427.127,352.039 427.288,352.11 427.449,352.307 427.611,352.609 427.772,352.986 \n  427.933,353.401 428.094,353.813 428.255,354.182 428.416,354.472 428.577,354.656 428.738,354.716 428.899,354.647 429.06,354.458 429.221,354.166 429.382,353.802 \n  429.543,353.401 429.704,353.002 429.865,352.645 430.026,352.365 430.187,352.187 430.348,352.128 430.51,352.195 430.671,352.378 430.832,352.66 430.993,353.013 \n  431.154,353.401 431.315,353.786 431.476,354.132 431.637,354.404 431.798,354.576 431.959,354.633 432.12,354.569 432.281,354.391 432.442,354.118 432.603,353.777 \n  432.764,353.401 432.925,353.027 433.086,352.692 433.247,352.429 433.409,352.262 433.57,352.207 433.731,352.269 433.892,352.441 434.053,352.705 434.214,353.036 \n  434.375,353.401 434.536,353.763 434.697,354.088 434.858,354.343 435.019,354.506 435.18,354.559 435.341,354.499 435.502,354.332 435.663,354.076 435.824,353.754 \n  435.985,353.401 436.147,353.049 436.308,352.734 436.469,352.485 436.63,352.328 436.791,352.276 436.952,352.334 437.113,352.496 437.274,352.745 437.435,353.057 \n  437.596,353.401 437.757,353.742 437.918,354.049 438.079,354.29 438.24,354.443 438.401,354.494 438.562,354.437 438.723,354.28 438.884,354.038 439.046,353.735 \n  439.207,353.401 439.368,353.069 439.529,352.771 439.69,352.536 439.851,352.387 440.012,352.338 440.173,352.392 440.334,352.545 440.495,352.781 440.656,353.076 \n  440.817,353.401 440.978,353.724 441.139,354.014 441.3,354.242 441.461,354.387 441.622,354.435 441.783,354.382 441.945,354.233 442.106,354.004 442.267,353.717 \n  442.428,353.401 442.589,353.086 442.75,352.804 442.911,352.581 443.072,352.44 443.233,352.393 443.394,352.445 443.555,352.59 443.716,352.813 443.877,353.092 \n  444.038,353.401 444.199,353.707 444.36,353.983 444.521,354.2 444.683,354.337 444.844,354.383 445.005,354.333 445.166,354.191 445.327,353.974 445.488,353.701 \n  445.649,353.401 445.81,353.102 445.971,352.833 446.132,352.622 446.293,352.487 446.454,352.443 446.615,352.492 446.776,352.629 446.937,352.842 447.098,353.108 \n  447.259,353.401 447.42,353.692 447.582,353.954 447.743,354.161 447.904,354.292 448.065,354.336 448.226,354.288 448.387,354.154 448.548,353.946 448.709,353.687 \n  448.87,353.401 449.031,353.116 449.192,352.86 449.353,352.659 449.514,352.53 449.675,352.488 449.836,352.534 449.997,352.666 450.158,352.868 450.319,353.121 \n  450.481,353.401 450.642,353.679 450.803,353.929 450.964,354.126 451.125,354.251 451.286,354.293 451.447,354.247 451.608,354.119 451.769,353.921 451.93,353.674 \n  452.091,353.401 452.252,353.129 452.413,352.885 452.574,352.692 452.735,352.57 452.896,352.529 453.057,352.573 453.219,352.698 453.38,352.892 453.541,353.134 \n  453.702,353.401 453.863,353.667 454.024,353.905 454.185,354.094 454.346,354.214 454.507,354.254 454.668,354.21 454.829,354.088 454.99,353.899 455.151,353.662 \n  455.312,353.401 455.473,353.141 455.634,352.907 455.795,352.723 455.956,352.605 456.118,352.566 456.279,352.609 456.44,352.729 456.601,352.913 456.762,353.145 \n  456.923,353.401 457.084,353.655 457.245,353.884 457.406,354.064 457.567,354.179 457.728,354.218 457.889,354.176 458.05,354.059 458.211,353.878 458.372,353.651 \n  458.533,353.401 458.694,353.151 458.855,352.928 459.017,352.751 459.178,352.638 459.339,352.601 459.5,352.641 459.661,352.756 459.822,352.933 459.983,353.156 \n  460.144,353.401 460.305,353.645 460.466,353.864 460.627,354.037 460.788,354.148 460.949,354.185 461.11,354.145 461.271,354.032 461.432,353.859 461.593,353.641 \n  461.754,353.401 461.916,353.161 462.077,352.946 462.238,352.777 462.399,352.668 462.56,352.632 462.721,352.671 462.882,352.782 463.043,352.952 463.204,353.165 \n  463.365,353.401 463.526,353.635 463.687,353.846 463.848,354.013 464.009,354.119 464.17,354.154 464.331,354.116 464.492,354.008 464.654,353.841 464.815,353.632 \n  464.976,353.401 465.137,353.171 465.298,352.964 465.459,352.8 465.62,352.696 465.781,352.662 465.942,352.699 466.103,352.805 466.264,352.969 466.425,353.174 \n  466.586,353.401 466.747,353.627 466.908,353.829 467.069,353.99 467.23,354.092 467.391,354.126 467.553,354.089 467.714,353.985 467.875,353.825 468.036,353.623 \n  468.197,353.401 468.358,353.179 468.519,352.98 468.68,352.823 468.841,352.722 469.002,352.689 469.163,352.725 469.324,352.827 469.485,352.985 469.646,353.182 \n  469.807,353.401 469.968,353.618 470.129,353.814 470.29,353.968 470.452,354.067 470.613,354.1 470.774,354.064 470.935,353.964 471.096,353.809 471.257,353.615 \n  471.418,353.401 471.579,353.187 471.74,352.995 471.901,352.843 472.062,352.746 472.223,352.714 472.384,352.749 472.545,352.847 472.706,352.999 472.867,353.19 \n  473.028,353.401 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604.047,352.874 604.208,353.532 604.369,354.166 604.53,354.686 \n  604.691,355.018 604.852,355.112 605.013,354.953 605.174,354.561 605.335,353.987 605.496,353.311 605.657,352.627 605.818,352.032 605.979,351.611 606.14,351.425 \n  606.301,351.503 606.463,351.838 606.624,352.384 606.785,353.068 606.946,353.795 607.107,354.462 607.268,354.974 607.429,355.258 607.59,355.27 607.751,355.005 \n  607.912,354.498 608.073,353.816 608.234,353.055 608.395,352.319 608.556,351.715 608.717,351.328 608.878,351.217 609.039,351.4 609.2,351.856 609.362,352.523 \n  609.523,353.311 609.684,354.109 609.845,354.805 610.006,355.3 610.167,355.52 610.328,355.43 610.489,355.04 610.65,354.399 610.811,353.596 610.972,352.741 \n  611.133,351.954 611.294,351.348 611.455,351.011 611.616,350.994 611.777,351.305 611.938,351.904 612.1,352.711 612.261,353.615 612.422,354.49 612.583,355.211 \n  612.744,355.675 612.905,355.811 613.066,355.595 613.227,355.053 613.388,354.255 613.549,353.311 613.71,352.352 613.871,351.513 614.032,350.914 614.193,350.646 \n  614.354,350.75 614.515,351.219 614.676,351.992 614.837,352.965 614.999,354.004 615.16,354.962 615.321,355.704 615.482,356.12 615.643,356.145 615.804,355.769 \n  615.965,355.037 616.126,354.048 616.287,352.936 616.448,351.856 616.609,350.962 616.77,350.384 616.931,350.21 617.092,350.472 617.253,351.142 617.414,352.133 \n  617.575,353.311 617.736,354.512 617.898,355.568 618.059,356.325 618.22,356.67 618.381,356.546 618.542,355.96 618.703,354.985 618.864,353.751 619.025,352.428 \n  619.186,351.201 619.347,350.245 619.508,349.704 619.669,349.663 619.83,350.139 619.991,351.076 620.152,352.354 620.313,353.799 620.474,355.209 620.636,356.385 \n  620.797,357.152 620.958,357.393 621.119,357.059 621.28,356.184 621.441,354.876 621.602,353.311 621.763,351.703 621.924,350.281 622.085,349.25 622.246,348.769 \n  622.407,348.921 622.568,349.703 622.729,351.022 622.89,352.706 623.051,354.528 623.212,356.233 623.373,357.574 623.535,358.349 623.696,358.43 623.857,357.783 \n  624.018,356.476 624.179,354.673 624.34,352.613 624.501,350.58 624.662,348.865 624.823,347.723 624.984,347.339 625.145,347.794 625.306,349.057 625.467,350.979 \n  625.628,353.311 625.789,355.738 625.95,357.919 626.111,359.532 626.272,360.321 626.434,360.14 626.595,358.969 626.756,356.931 626.917,354.275 627.078,351.351 \n  627.239,348.563 627.4,346.315 627.561,344.958 627.722,344.732 627.883,345.732 628.044,347.884 628.205,350.947 628.366,354.538 628.527,358.175 628.688,361.342 \n  628.849,363.555 629.01,364.431 629.171,363.753 629.333,361.503 629.494,357.886 629.655,353.311 629.816,348.355 629.977,343.694 630.138,340.021 630.299,337.954 \n  630.46,337.945 630.621,340.207 630.782,344.663 630.943,350.929 631.104,358.333 631.265,365.973 631.426,372.798 631.587,377.726 631.748,379.761 631.909,378.113 \n  632.071,372.304 632.232,362.24 632.393,348.249 632.554,331.073 632.715,311.816 632.876,291.849 633.037,272.689 633.198,255.854 633.359,242.714 633.52,234.361 \n  633.681,231.496 633.842,234.361 634.003,242.714 634.164,255.854 634.325,272.689 634.486,291.849 634.647,311.816 634.808,331.073 634.97,348.249 635.131,362.24 \n  635.292,372.304 635.453,378.113 635.614,379.761 635.775,377.726 635.936,372.798 636.097,365.973 636.258,358.333 636.419,350.929 636.58,344.663 636.741,340.207 \n  636.902,337.945 637.063,337.954 637.224,340.021 637.385,343.694 637.546,348.355 637.707,353.311 637.869,357.886 638.03,361.503 638.191,363.753 638.352,364.431 \n  638.513,363.555 638.674,361.342 638.835,358.175 638.996,354.538 639.157,350.947 639.318,347.884 639.479,345.732 639.64,344.732 639.801,344.958 639.962,346.315 \n  640.123,348.563 640.284,351.351 640.445,354.275 640.607,356.931 640.768,358.969 640.929,360.14 641.09,360.321 641.251,359.532 641.412,357.919 641.573,355.738 \n  641.734,353.311 641.895,350.979 642.056,349.057 642.217,347.794 642.378,347.339 642.539,347.723 642.7,348.865 642.861,350.58 643.022,352.613 643.183,354.673 \n  643.344,356.476 643.506,357.783 643.667,358.43 643.828,358.349 643.989,357.574 644.15,356.233 644.311,354.528 644.472,352.706 644.633,351.022 644.794,349.703 \n  644.955,348.921 645.116,348.769 645.277,349.25 645.438,350.281 645.599,351.703 645.76,353.311 645.921,354.876 646.082,356.184 646.243,357.059 646.405,357.393 \n  646.566,357.152 646.727,356.385 646.888,355.209 647.049,353.799 647.21,352.354 647.371,351.076 647.532,350.139 647.693,349.663 647.854,349.704 648.015,350.245 \n  648.176,351.201 648.337,352.428 648.498,353.751 648.659,354.985 648.82,355.96 648.981,356.546 649.142,356.67 649.304,356.325 649.465,355.568 649.626,354.512 \n  649.787,353.311 649.948,352.133 650.109,351.142 650.27,350.472 650.431,350.21 650.592,350.384 650.753,350.962 650.914,351.856 651.075,352.936 651.236,354.048 \n  651.397,355.037 651.558,355.769 651.719,356.145 651.88,356.12 652.042,355.704 652.203,354.962 652.364,354.004 652.525,352.965 652.686,351.992 652.847,351.219 \n  653.008,350.75 653.169,350.646 653.33,350.914 653.491,351.513 653.652,352.352 653.813,353.311 653.974,354.255 654.135,355.053 654.296,355.595 654.457,355.811 \n  654.618,355.675 654.779,355.211 654.941,354.49 655.102,353.615 655.263,352.711 655.424,351.904 655.585,351.305 655.746,350.994 655.907,351.011 656.068,351.348 \n  656.229,351.954 656.39,352.741 656.551,353.596 656.712,354.399 656.873,355.04 657.034,355.43 657.195,355.52 657.356,355.3 657.517,354.805 657.678,354.109 \n  657.84,353.311 658.001,352.523 658.162,351.856 658.323,351.4 658.484,351.217 658.645,351.328 658.806,351.715 658.967,352.319 659.128,353.055 659.289,353.816 \n  659.45,354.498 659.611,355.005 659.772,355.27 659.933,355.258 660.094,354.974 660.255,354.462 660.416,353.795 660.578,353.068 660.739,352.384 660.9,351.838 \n  661.061,351.503 661.222,351.425 661.383,351.611 661.544,352.032 661.705,352.627 661.866,353.311 662.027,353.987 662.188,354.561 662.349,354.953 662.51,355.112 \n  662.671,355.018 662.832,354.686 662.993,354.166 663.154,353.532 663.315,352.874 663.477,352.284 663.638,351.844 663.799,351.614 663.96,351.623 664.121,351.867 \n  664.282,352.311 664.443,352.89 664.604,353.522 664.765,354.117 664.926,354.594 665.087,354.887 665.248,354.956 665.409,354.795 665.57,354.428 665.731,353.909 \n  665.892,353.311 666.053,352.719 666.214,352.216 666.376,351.871 666.537,351.73 666.698,351.812 666.859,352.102 667.02,352.559 667.181,353.116 667.342,353.695 \n  667.503,354.215 667.664,354.604 667.825,354.808 667.986,354.801 668.147,354.586 668.308,354.194 668.469,353.683 668.63,353.124 668.791,352.597 668.952,352.174 \n  669.114,351.914 669.275,351.852 669.436,351.994 669.597,352.319 669.758,352.78 669.919,353.311 670.08,353.837 670.241,354.285 670.402,354.593 670.563,354.719 \n  670.724,354.647 670.885,354.389 671.046,353.982 671.207,353.485 671.368,352.967 671.529,352.503 671.69,352.155 671.851,351.972 672.013,351.977 672.174,352.169 \n  672.335,352.52 672.496,352.977 672.657,353.478 672.818,353.951 672.979,354.331 673.14,354.565 673.301,354.622 673.462,354.495 673.623,354.203 673.784,353.789 \n  673.945,353.311 674.106,352.837 674.267,352.433 674.428,352.155 674.589,352.041 674.75,352.106 674.912,352.338 675.073,352.705 675.234,353.154 675.395,353.621 \n  675.556,354.041 675.717,354.356 675.878,354.522 676.039,354.518 676.2,354.344 676.361,354.028 676.522,353.613 676.683,353.159 676.844,352.73 677.005,352.385 \n  677.166,352.173 677.327,352.121 677.488,352.236 677.649,352.501 677.811,352.877 677.972,353.311 678.133,353.742 678.294,354.109 678.455,354.363 678.616,354.467 \n  678.777,354.408 678.938,354.197 679.099,353.863 679.26,353.454 679.421,353.028 679.582,352.644 679.743,352.357 679.904,352.205 680.065,352.209 680.226,352.367 \n  680.387,352.656 680.549,353.035 680.71,353.45 680.871,353.842 681.032,354.158 681.193,354.353 681.354,354.4 681.515,354.295 681.676,354.053 681.837,353.709 \n  681.998,353.311 682.159,352.916 682.32,352.578 682.481,352.346 682.642,352.25 682.803,352.303 682.964,352.497 683.125,352.804 683.286,353.179 683.448,353.571 \n  683.609,353.924 683.77,354.188 683.931,354.328 684.092,354.325 684.253,354.18 684.414,353.914 684.575,353.565 684.736,353.183 684.897,352.821 685.058,352.53 \n  685.219,352.351 685.38,352.306 685.541,352.403 685.702,352.626 685.863,352.944 686.024,353.311 686.185,353.676 686.347,353.987 686.508,354.202 686.669,354.291 \n  686.83,354.242 686.991,354.063 687.152,353.78 687.313,353.433 687.474,353.07 687.635,352.744 687.796,352.499 687.957,352.369 688.118,352.372 688.279,352.506 \n  688.44,352.752 688.601,353.075 688.762,353.429 688.923,353.765 689.085,354.035 689.246,354.202 689.407,354.243 689.568,354.153 689.729,353.946 689.89,353.652 \n  690.051,353.311 690.212,352.972 690.373,352.682 690.534,352.483 690.695,352.4 690.856,352.445 691.017,352.612 691.178,352.875 691.339,353.198 691.5,353.535 \n  691.661,353.838 691.822,354.067 691.984,354.188 692.145,354.185 692.306,354.06 692.467,353.831 692.628,353.53 692.789,353.201 692.95,352.888 693.111,352.636 \n  693.272,352.48 693.433,352.442 693.594,352.525 693.755,352.718 693.916,352.993 694.077,353.311 694.238,353.627 694.399,353.898 694.56,354.084 694.721,354.162 \n  694.883,354.12 695.044,353.964 695.205,353.718 695.366,353.417 695.527,353.102 695.688,352.818 695.849,352.604 696.01,352.491 696.171,352.493 696.332,352.61 \n  696.493,352.824 696.654,353.105 696.815,353.414 696.976,353.707 697.137,353.943 697.298,354.089 697.459,354.125 697.62,354.047 697.782,353.867 697.943,353.609 \n  698.104,353.311 698.265,353.014 698.426,352.761 698.587,352.585 698.748,352.513 698.909,352.552 699.07,352.698 699.231,352.928 699.392,353.212 699.553,353.507 \n  699.714,353.774 699.875,353.975 700.036,354.081 700.197,354.079 700.358,353.97 700.52,353.769 700.681,353.504 700.842,353.214 701.003,352.938 701.164,352.717 \n  701.325,352.579 701.486,352.545 701.647,352.618 701.808,352.788 701.969,353.03 702.13,353.311 702.291,353.59 702.452,353.829 702.613,353.994 702.774,354.063 \n  702.935,354.026 703.096,353.889 703.257,353.671 703.419,353.404 703.58,353.126 703.741,352.874 703.902,352.685 704.063,352.585 704.224,352.586 704.385,352.689 \n  704.546,352.879 704.707,353.129 704.868,353.403 705.029,353.662 705.19,353.872 705.351,354.001 705.512,354.034 705.673,353.965 705.834,353.805 705.995,353.576 \n  706.156,353.311 706.318,353.047 706.479,352.822 706.64,352.665 706.801,352.601 706.962,352.635 707.123,352.765 707.284,352.97 707.445,353.222 707.606,353.486 \n  707.767,353.724 707.928,353.903 708.089,353.998 708.25,353.996 708.411,353.899 708.572,353.719 708.733,353.483 708.894,353.224 709.056,352.978 709.217,352.78 \n  709.378,352.657 709.539,352.626 709.7,352.691 709.861,352.843 710.022,353.06 710.183,353.311 710.344,353.561 710.505,353.775 710.666,353.923 710.827,353.984 \n  710.988,353.951 711.149,353.828 711.31,353.634 711.471,353.395 711.632,353.145 711.793,352.919 711.955,352.75 712.116,352.659 712.277,352.66 712.438,352.753 \n  712.599,352.923 712.76,353.147 712.921,353.393 713.082,353.627 713.243,353.815 713.404,353.932 713.565,353.961 713.726,353.899 713.887,353.755 714.048,353.549 \n  714.209,353.311 714.37,353.073 714.531,352.87 714.692,352.73 714.854,352.671 715.015,352.702 715.176,352.819 715.337,353.004 715.498,353.231 715.659,353.469 \n  715.82,353.683 715.981,353.845 716.142,353.931 716.303,353.93 716.464,353.842 716.625,353.68 716.786,353.467 716.947,353.233 717.108,353.01 717.269,352.831 \n  717.43,352.72 717.591,352.692 717.753,352.751 717.914,352.888 718.075,353.084 718.236,353.311 718.397,353.537 718.558,353.731 718.719,353.865 718.88,353.921 \n  719.041,353.891 719.202,353.78 719.363,353.603 719.524,353.387 719.685,353.16 719.846,352.956 720.007,352.802 720.168,352.72 720.329,352.721 720.491,352.805 \n  720.652,352.959 720.813,353.162 720.974,353.386 721.135,353.598 721.296,353.769 721.457,353.875 721.618,353.902 721.779,353.845 721.94,353.715 722.101,353.528 \n  722.262,353.311 722.423,353.095 722.584,352.91 722.745,352.782 722.906,352.729 723.067,352.757 723.228,352.863 723.39,353.031 723.551,353.238 723.712,353.455 \n  723.873,353.65 724.034,353.797 724.195,353.876 724.356,353.875 724.517,353.795 724.678,353.647 724.839,353.453 725,353.239 725.161,353.037 725.322,352.873 \n  725.483,352.772 725.644,352.746 725.805,352.8 725.966,352.925 726.127,353.104 726.289,353.311 726.45,353.517 726.611,353.694 726.772,353.817 726.933,353.868 \n  727.094,353.841 727.255,353.739 727.416,353.578 727.577,353.38 727.738,353.173 727.899,352.986 728.06,352.845 728.221,352.77 728.382,352.771 728.543,352.848 \n  728.704,352.989 728.865,353.175 729.027,353.379 729.188,353.573 729.349,353.73 729.51,353.827 729.671,353.852 729.832,353.801 729.993,353.681 730.154,353.509 \n  730.315,353.311 \n  \"/>\n</svg>\n"
          },
          "execution_count": 3,
          "metadata": {},
          "output_type": "execute_result"
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### Dirichlet積分の公式\n\n条件収束する広義積分に関する公式\n\n$$\n\\ds\\int_0^\\infty\\frac{\\sin t}{t}\\,dt=\\frac{\\pi}{2}\n$$\n\nより\n\n$$\n\\int_{-\\infty}^\\infty D_N(x)\\,dx = \\frac{2}{\\pi}\\int_0^\\infty \\frac{\\sin(2\\pi Nx)}{x}\\,dx =\n\\frac{2}{\\pi}\\int_0^\\infty \\frac{\\sin(2\\pi Nx)}{x}\\,dx = \\frac{2}{\\pi}\\frac{\\pi}{2}=1.\n$$\n\n三番目の等号で $x=t/(2\\pi N)$ と置換して計算した.  この公式は条件収束する積分の公式であることに注意せよ.\n\n上の公式(**Dirichlet積分の公式**と呼ばれる)については\n\n* <a href=\"http://nbviewer.jupyter.org/github/genkuroki/Calculus/blob/master/09%20integration.ipynb\">09 積分</a>\n\nの<a href=\"http://nbviewer.jupyter.org/github/genkuroki/Calculus/blob/master/09%20integration.ipynb#Dirichlet積分とその一般化\">Dirichlet積分とその一般化</a>の解説を参照せよ."
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "**Dirichlet積分の公式補足:** 条件収束する広義積分に関する次の公式が成立している:\n\n$$\n\\int_{-\\infty}^\\infty \\frac{\\sin(ax)}{\\pi x}\\,dx = \\sign(a) =\n\\begin{cases}\n1 & (a<0) \\\\\n0 & (a=0) \\\\\n-1 & (a>0).\n\\end{cases}\n$$\n\n**証明:** $a=0$ のとき, 被積分函数が恒等的に $0$ になるので, 積分も $0$ になる. $a\\ne 0$ と仮定する.\n\n$$\n\\int_{-\\infty}^\\infty \\frac{\\sin(ax)}{\\pi x}\\,dx = \\frac{2}{\\pi}\\int_0^\\infty \\frac{\\sin(ax)}{x}\\,dx\n$$\n\nであるから, $x = t/|a|$ と置換すると, $a/|a|=\\sign(a)$ なので,\n\n$$\n\\frac{2}{\\pi}\\int_0^\\infty \\frac{\\sin(ax)}{x}\\,dx =\n\\frac{2}{\\pi}\\int_0^\\infty \\frac{\\sign(a)\\sin(t)}{t}\\,dt = \\sign(a)\\frac{2}{\\pi}\\frac{\\pi}{2}=\\sign(a).\n\\qquad\\QED\n$$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### Riemannの局所性定理\n\n**Riemannの局所性定理:** $f$ は $\\R$ 上の $L^1$ 函数であるとし, $\\delta>0$ であるとする. このとき,\n\n$$\n\\lim_{N\\to\\infty}\\int_{-\\infty}^{-\\delta} f(x+y)D_N(y)\\,dy = 0, \\quad\n\\lim_{N\\to\\infty}\\int_\\delta^\\infty f(x+y)D_N(y)\\,dy = 0.\n$$\n\nこれより, $\\int_{-N}^N \\hat{f}(p)e^{2\\pi ipx}\\,dp = \\int_\\R f(x+y) D_N(y)\\,dy$ が $N\\to\\infty$ で収束することと, $\\int_{-\\delta}^\\delta f(x+y) D_N(y)\\,dy$ が $N\\to\\infty$ で収束することは同値であり, 収束する場合には同じ値に収束することがわかる.  後者が収束するか否かは $x$ の近くでの函数 $f$ の様子だけで決まることに注意せよ. \n\n**証明:**\n$$\n\\begin{aligned}\n&\n\\int_{-\\infty}^{-\\delta} f(x+y)D_N(y)\\,dy = \\int_{-\\infty}^{-\\delta} \\frac{f(x+y)}{y} \\sin(2\\pi Ny)\\,dy,\n\\\\ &\n\\int_\\delta^\\infty f(x+y)D_N(y)\\,dy = \\int_{-\\infty}^{-\\delta} \\frac{f(x+y)}{y} \\sin(2\\pi Ny)\\,dy.\n\\end{aligned}\n$$\n\nであり, 区間 $(-\\infty,-\\delta)$ と $(\\delta,\\infty)$ のそれぞれの上で函数 $f(x+y)/y$ は $L^1$ 函数なので, Riemann-Lebesgueの定理より, これらは $N\\to\\infty$ で $0$ に収束する. $\\QED$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### Fourier変換の逆変換の収束性 (Diniの条件)"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "**補題:** $\\delta>0$ のとき,\n\n$$\n\\lim_{N\\to\\infty}\\int_{-\\infty}^{-\\delta} D_N(y)dy = 0, \\quad\n\\lim_{N\\to\\infty}\\int_\\delta^\\infty D_N(y)dy = 0. \n$$\n\n**注意:** 条件収束する広義積分の極限に関する結果なのでRiemann-Lebesgueの定理を直接使用できない. $\\QED$\n\n**証明:** $D_N(y)$ は偶函数なので後者のみを示せば十分である. Dirichlet積分の公式より $\\ds\\lim_{N\\to\\infty}\\int_0^{2\\pi N\\delta} \\frac{\\sin t}{t}\\,dt=\\frac{\\pi}{2}$ なので, $\\ds\\lim_{N\\to\\infty}\\int_{2\\pi N\\delta}^\\infty \\frac{\\sin t}{\\pi t}\\,dt=0$ となることがわかる. ゆえに, $x=t/(2\\pi N)$ とおくと, \n\n$$\n\\int_\\delta^\\infty D_N(y)dy = \\int_\\delta^\\infty \\frac{\\sin(2\\pi Nx)}{\\pi x}\\,dx =\n\\int_{2\\pi N\\delta}^\\infty \\frac{\\sin t}{\\pi t}\\,dt\\to 0 \\quad (N\\to\\infty).\n\\qquad\\QED\n$$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "**定理(Diniの条件):** $f$ は $\\R$ 上の $L^1$ 函数であり, $x\\in \\R$ であるとする.とする. さらに, ある $\\delta>0$ が存在して, \n\n$$\n\\int_0^\\delta \\frac{|(f(x+y) + f(x-y) - 2f(x)|}{y}\\,dy <\\infty\n\\tag{$*$}\n$$\n\nが成立していると仮定する. この条件($*$)を**Diniの条件**と呼ぶ. このとき, \n\n$$\n\\lim_{N\\to\\infty}\\int_{-N}^N \\hat{f}(p)e^{2\\pi ipx}\\,dp = \\lim_{N\\to\\infty}\\int_\\R f(x+y) D_N(y)\\,dy = f(x).\n$$\n\n**証明:** $\\int_{-N}^N \\hat{f}(p)e^{2\\pi ipx}\\,dp - f(x)$ が $N\\to\\infty$ で $0$ に収束することを示せばよい. \n\n$$\n\\begin{aligned}\n\\int_{-N}^N \\hat{f}(p)e^{2\\pi ipx}\\,dp - f(x) &=\n\\int_\\R f(x+y) D_N(y)\\,dy - f(x) \\int_{-\\infty}^\\infty D_N(y)\\,dy\n\\\\ &=\n\\int_{-\\infty}^\\infty (f(x+y)-f(x)) D_N(y)\\,dy\n\\\\ &=\n\\int_0^\\infty (f(x+y)-f(x)) D_N(y)\\,dy +\n\\int_{-\\infty}^0 (f(x+y)-f(x)) D_N(y)\\,dy\n\\\\ &=\n\\int_0^\\infty (f(x+y)-f(x)) D_N(y)\\,dy +\n\\int_0^\\infty (f(x-y)-f(x)) D_N(y)\\,dy\n\\\\ &=\n\\int_0^\\infty (f(x+y)+f(x-y)-2f(x)) D_N(y)\\,dy\n\\\\ &=\n\\int_0^\\infty \\frac{f(x+y)+f(x-y)-2f(x)}{y} \\frac{\\sin(2\\pi N y)}{\\pi}\\,dy = I_N + J_N.\n\\end{aligned}\n$$\n\nここで, \n\n$$\n\\delta > 0, \\quad\nI_N = \\int_0^\\delta \\frac{f(x+y)+f(x-y)-2f(x)}{y} \\frac{\\sin(2\\pi N y)}{\\pi}\\,dy, \\quad\nJ_N = \\int_\\delta^\\infty \\frac{f(x+y)+f(x-y)-2f(x)}{y} \\frac{\\sin(2\\pi N y)}{\\pi}\\,dy.\n$$\n\n$N\\to\\infty$ のとき, Riemannの局所性定理と上の補題より $J_N\\to 0$ となり, Diniの条件の仮定とRiemann-Lebesgueの定理より $I_N\\to 0$ となる. これで示したいことが示せた. $\\QED$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "**例:** $f$ は $\\R$ 上の $L^1$ 函数であるとする. $y\\searrow 0$ のとき $f(x+y), f(x-y)$ は収束していると仮定し, それぞれの収束先を $f(x+0), f(x-0)$ と書くことにする. さらに $y\\searrow 0$ で $\\ds\\frac{f(x+y)-f(x+0)}{y}$ と $\\ds\\frac{f(x-y)-f(x-0)}{y}$ は収束していると仮定する. \n\nこのとき, 十分小さな $\\delta>0$ を取ると, $\\ds\\frac{f(x+y)+f(x-y)-(f(x+0)+f(x-0))}{y}$ は $0<y<\\delta$ で有界になり, 特にそこで可積分になる. ゆえに $f(x)=(f(x+0)+f(x-0))/2$ ならば, Diniの条件が満たされており, 上の定理より,\n\n$$\n\\lim_{N\\to\\infty}\\int_{-N}^N \\hat{f}(p)e^{2\\pi ipx}\\,dp = \\lim_{N\\to\\infty}\\int_\\R f(x+y) D_N(y)\\,dy = f(x).\n$$\n\n特に $f$ が $x$ で微分可能ならばこれが成立している. さらにもしもFourier変換 $\\hat{f}(p)$ が可積分でかつDiniの条件が満たされているならば,\n\n$$\n\\hat{f}(p)=\\int_{-\\infty}^\\infty f(x)e^{-2\\pi ipx}\\,dx\n\\implies\n\\int_{-\\infty}^\\infty \\hat{f}(p)e^{2\\pi ipx}\\,dp = f(x)\n$$\n\nが絶対収束する積分の公式として成立している. $\\QED$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "**例:** $a>0$ であるとし, $f(x)$ を\n\n$$\nf(x) = \\begin{cases}\n1 & (-a<x<a) \\\\\n1/2 & (a=\\pm a) \\\\\n0 & (\\text{otherwise})\n\\end{cases}\n$$\n\nと定める. このとき, Diniの条件が満たされているので, \n\n$$\n\\lim_{N\\to\\infty}\\int_{-N}^N \\hat{f}(p)e^{2\\pi ipx}\\,dp = f(x)\n\\tag{$*$}\n$$\n\nが成立している. これを直接の計算で確認してみよう.\n\n$$\n\\hat{f}(p) = \\int_{-a}^a e^{-2\\pi ipx}\\,dx = \\frac{e^{-2\\pi ipa}-e^{2\\pi ipa}}{-2\\pi ip} =\n\\frac{\\sin(2\\pi a p)}{\\pi p}.\n$$\n\nゆえに, \n\n$$\n\\begin{aligned}\n\\int_{-N}^N \\hat{f}(p)e^{2\\pi ipx}\\,dp &=\n\\int_{-N}^N \\frac{\\sin(2\\pi ap)}{\\pi p} e^{2\\pi ipx}\\,dp =\n\\int_{-N}^N \\frac{\\sin(2\\pi ap)\\cos(2\\pi xp)}{\\pi p}\\,dp\n\\\\ &=\n\\frac{1}{2}\\int_{-N}^N \\frac{\\sin(2\\pi(a+x)p)+\\sin(2\\pi(a-x)p)}{\\pi p}\\,dp.\n\\end{aligned}\n$$\n\n2つ目の等号で $\\ds e^{2\\pi ipx}=\\cos(2\\pi xp)+i\\sin(2\\pi xp)$ と $\\ds\\frac{\\sin(2\\pi a p)\\sin(2\\pi xp)}{\\pi p}$ が $p$ の奇函数であることを用いた. 3つ目の等号では三角函数の加法公式を使って示される $2\\sin\\alpha\\,\\cos\\beta = \\sin(\\alpha+\\beta) + \\sin(\\alpha-\\beta)$ を用いた.\n\nそして, Dirichlet積分の公式より, \n\n$$\n\\begin{aligned}\n&\n\\lim_{N\\to\\infty}\\frac{1}{2}\\int_{-N}^N \\frac{\\sin(2\\pi(a+x)p)}{\\pi p}\\,dp =\n\\begin{cases}\n1/2 & (x>-a) \\\\\n0 & (x=a) \\\\\n-1/2 & (x<-a),\n\\end{cases}\n\\\\ &\n\\lim_{N\\to\\infty}\\frac{1}{2}\\int_{-N}^N \\frac{\\sin(2\\pi(a-x)p)}{\\pi p}\\,dp =\n\\begin{cases}\n1/2 & (x<a) \\\\\n0 & (x=a) \\\\\n-1/2 & (x>a).\n\\end{cases}\n\\end{aligned}\n$$\n\n以上を合わせると, 式($*$)が成立していることがわかる. $\\QED$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "## Fourier級数の収束\n\nFourier変換の逆変換の収束の議論とほぼ同じ議論を以下では行う."
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### Fourier級数の定義\n\n$f(x)$ は周期 $1$ を持つ $\\R$ 上の函数であると仮定する: $f(x+1)=f(x)$.  このとき, 整数 $k\\in\\Z$ に対して, $f(x)$ のFourier係数 $a_k(f)$ を\n\n$$\na_k(f) = \\int_0^1 f(x)e^{-2\\pi i kx}\\,dx\n$$\n\nと定める. そして, \n\n$$\n\\sum_{k\\in\\Z} a_k(f) e^{2\\pi ikx}\n$$\n\nを $f(x)$ の**Fourier級数展開**と呼ぶ. "
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### Fourier展開のDirichlet核\n\n**Dirichlet核** $D_N(x)$ を\n\n$$\nD_N(x) = \\sum_{k=-N}^N e^{2\\pi ikx} = e^{-2\\pi iNx}\\frac{e^{2\\pi i(2N+1)x}-1}{e^{2\\pi i x}-1} =\n\\frac{\\sin(\\pi(2N+1)x)}{\\sin(\\pi x)}\n$$\n\nと定める. ただし, $x\\in\\Z$ のときには $D_N(x)=2N+1$ と定めておく.  $D_N(x)$ は有界な偶函数で周期 $1$ を持つ.\n\nこのとき, \n\n$$\n\\begin{aligned}\n\\sum_{k=-N}^N a_k(f)e^{2\\pi ikx} &=\n\\sum_{k=-N}^N \\left(\\int_0^1 f(y)e^{2\\pi i ky}\\,dy\\right)e^{2\\pi ikx} =\n\\int_0^1 f(y)\\left(\\sum_{k=-N}^N e^{2\\pi i k(x-y)}\\,dx\\right)\\,dy \n\\\\ &=\n\\int_0^1 f(y)D_N(x-y)\\,dy =\n\\int_{-1/2}^{1/2} f(x+y)D_N(y)\\,dy.\n\\end{aligned}\n$$\n\n最後の等号で $y$ を $x+y$ で置換し, $D_N(-y)=D_N(y)$ と $f(x+y)D_N(y)$ が $y$ について周期 $1$ を持つ函数になることを用いた.\n\n$$\n\\int_0^1 D_N(y)\\,dy = \\sum_{k=-N}^N\\int_0^1 e^{2\\pi iky} \\,dy = \\int_0^1 1 \\,dy = 1.\n$$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "**問題:** Fourier級数のDirichlet核のグラフを描け. $\\QED$\n\n**解説:** 上の $D_N(x)$ は $D_N(x+1)=D_N(x)$ を満たしているので, $-1/2\\leqq x\\leqq 1/2$ でグラフを描けば十分である. $D_N(x)$ のグラフは $N$ 個の山を持ち, 振動が細かくなり, $N$ が大きくなると, $x=0$ の近くでは(より正確には整数の近くの $x$ においては) $D_N(x)$ の値が大きくなり, 他での値は小さくなる.\n\n次のセルを見よ. $\\QED$"
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "# フーリエ級数のディリクレ核 D_N(x) のグラフ\n\nD(N,x) = iszero(x) ? 2N+1 : sin(π*(2N+1)*x)/(sin(π*x))\nPP = []\nfor N in [1:6; 10; 20; 30]\n    x = -0.5:0.001:0.5\n    P = plot(x, D.(N,x), title=\"N=$N\", titlefontsize=10)\n    push!(PP, P)\nend\nplot(PP..., size=(750, 600), legend=false)",
      "execution_count": 4,
      "outputs": [
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      "source": "**問題:** Fourier級数のDirichlet核 $\\ds D_N(x)=\\frac{\\sin(\\pi(2N+1)x)}{\\sin(\\pi x)}$ は上の方で示したように, $\\int_0^1 D_N(x)\\,dx=1$ を満たしている. しかし, その絶対値 $|D_N(x)|$ の積分については,\n\n$$\n\\int_0^1 |D_N(x)|\\,dx \\to \\infty \\quad (N\\to\\infty)\n$$\n\nが成立していることを示せ.\n\n**解答例:** $|\\sin(\\pi x)| \\leqq \\pi |x|$ なので,\n\n$$\n\\int_0^1 |D_N(x)|\\,dx = 2\\int_0^{1/2} |D_N(x)|\\,dx \\geqq\n2\\int_0^{1/2} \\frac{|\\sin(\\pi(2N+1)x)|}{\\pi x}\\,dx.\n$$\n\n$x=t/(π(2N+1))$ と置換すると, \n\n$$\n\\begin{aligned}\n&\n2\\int_0^{1/2} \\frac{|\\sin(\\pi(2N+1)x)|}{\\pi x}\\,dx =\n\\frac{2}{\\pi}\\int_0^{\\pi(2N+1)/2} \\frac{|\\sin t|}{t}\\,dt =\n\\frac{2}{\\pi}\\sum_{k=1}^{2N+1}\\int_{(k-1)\\pi/2}^{k\\pi/2} \\frac{|\\sin t|}{t}\\,dt\n\\\\ &\\qquad\\geqq\n\\frac{2}{\\pi}\\sum_{k=1}^{2N+1}\\int_{(k-1)\\pi/2}^{k\\pi/2}\\frac{|\\sin t|}{k\\pi/2}\\,dt =\n\\left(\\frac{2}{\\pi}\\right)^2\\int_0^{\\pi/2}\\sin t\\,dt \\sum_{k=1}^{2N+1}\\frac{1}{k} =\n\\left(\\frac{2}{\\pi}\\right)^2\\sum_{k=1}^{2N+1}\\frac{1}{k}.\n\\end{aligned}\n$$\n\nそして, $\\ds\\sum_{k=1}^{2N+1}\\frac{1}{k}\\sim\\log(2N+1)=O(\\log N)$. 以上をまとめると,\n\n$$\n\\int_0^1 |D_N(x)|\\,dx \\geqq \\left(\\frac{2}{\\pi}\\right)^2\\sum_{k=1}^{2N+1}\\frac{1}{k} = O(\\log N).\n$$\n\nこれより, 左辺は $N\\to\\infty$ のとき $\\log N$ 以上のオーダーで無限大に発散することがわかる. $\\QED$\n\n**解説:** $N\\to \\infty$ のとき, $\\ds \\int_0^1 D_N(x)\\,dx=1$ の左辺の積分はそれぞれ $\\infty,-\\infty$ に発散する正と負の成分が互いにキャンセルすることによって有限の値が残るという計算になる. これが, Fourier級数の収束の理論がややこしくなる理由であると考えられる. $\\QED$"
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      "source": "# |sin πx| ≤ π|x| の確認\n\nf(x) = abs(sin(π*x))\ng(x) = π*abs(x)\nx = -1/2:0.002:1/2\nplot(size=(500, 350), legend=:top)\nplot!(x, f.(x), label=\"|sin(pi x)|\")\nplot!(x, g.(x), label=\"pi |x|\")",
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364.772,173 365.668,171.75 366.563,170.501 367.459,169.251 368.355,168.002 369.251,166.752 370.146,165.502 371.042,164.253 371.938,163.003 \n  372.833,161.754 373.729,160.504 374.625,159.255 375.52,158.005 376.416,156.756 377.312,155.506 378.207,154.257 379.103,153.007 379.999,151.758 380.894,150.508 \n  381.79,149.259 382.686,148.009 383.581,146.76 384.477,145.51 385.373,144.261 386.268,143.011 387.164,141.762 388.06,140.512 388.956,139.262 389.851,138.013 \n  390.747,136.763 391.643,135.514 392.538,134.264 393.434,133.015 394.33,131.765 395.225,130.516 396.121,129.266 397.017,128.017 397.912,126.767 398.808,125.518 \n  399.704,124.268 400.599,123.019 401.495,121.769 402.391,120.52 403.286,119.27 404.182,118.021 405.078,116.771 405.973,115.522 406.869,114.272 407.765,113.022 \n  408.66,111.773 409.556,110.523 410.452,109.274 411.348,108.024 412.243,106.775 413.139,105.525 414.035,104.276 414.93,103.026 415.826,101.777 416.722,100.527 \n  417.617,99.2777 418.513,98.0282 419.409,96.7787 420.304,95.5291 421.2,94.2796 422.096,93.0301 422.991,91.7806 423.887,90.531 424.783,89.2815 425.678,88.032 \n  426.574,86.7825 427.47,85.5329 428.365,84.2834 429.261,83.0339 430.157,81.7844 431.053,80.5348 431.948,79.2853 432.844,78.0358 433.74,76.7863 434.635,75.5367 \n  435.531,74.2872 436.427,73.0377 437.322,71.7882 438.218,70.5387 439.114,69.2891 440.009,68.0396 440.905,66.7901 441.801,65.5406 442.696,64.291 443.592,63.0415 \n  444.488,61.792 445.383,60.5425 446.279,59.2929 447.175,58.0434 448.07,56.7939 448.966,55.5444 449.862,54.2948 450.757,53.0453 451.653,51.7958 452.549,50.5463 \n  453.445,49.2967 454.34,48.0472 455.236,46.7977 456.132,45.5482 457.027,44.2986 457.923,43.0491 458.819,41.7996 459.714,40.5501 460.61,39.3006 461.506,38.051 \n  462.401,36.8015 463.297,35.552 464.193,34.3025 465.088,33.0529 465.984,31.8034 466.88,30.5539 467.775,29.3044 468.671,28.0548 469.567,26.8053 470.462,25.5558 \n  471.358,24.3063 472.254,23.0567 473.15,21.8072 474.045,20.5577 474.941,19.3082 475.837,18.0586 476.732,16.8091 477.628,15.5596 478.524,14.3101 479.419,13.0605 \n  480.315,11.811 \n  \"/>\n<polygon clip-path=\"url(#clip6900)\" points=\"\n179.733,72.051 278.107,72.051 278.107,26.691 179.733,26.691 \n  \" fill=\"#ffffff\" fill-opacity=\"1\"/>\n<polyline clip-path=\"url(#clip6900)\" style=\"stroke:#000000; stroke-width:1; stroke-opacity:1; fill:none\" points=\"\n  179.733,72.051 278.107,72.051 278.107,26.691 179.733,26.691 179.733,72.051 \n  \"/>\n<polyline clip-path=\"url(#clip6900)\" style=\"stroke:#009af9; stroke-width:1; stroke-opacity:1; fill:none\" points=\"\n  184.733,41.811 214.733,41.811 \n  \"/>\n<g clip-path=\"url(#clip6900)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 219.733, 46.311)\" x=\"219.733\" y=\"46.311\">|sin(pi x)|</text>\n</g>\n<polyline clip-path=\"url(#clip6900)\" style=\"stroke:#e26f46; stroke-width:1; stroke-opacity:1; fill:none\" points=\"\n  184.733,56.931 214.733,56.931 \n  \"/>\n<g clip-path=\"url(#clip6900)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 219.733, 61.431)\" x=\"219.733\" y=\"61.431\">pi |x|</text>\n</g>\n</svg>\n"
          },
          "execution_count": 5,
          "metadata": {},
          "output_type": "execute_result"
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "# |D_N(x)| の積分の漸近挙動の確認\n\nD(N,x) = iszero(x) ? π*(2N+1) : sin(π*(2N+1)*x)/sin(π*x)\nF(N) = quadgk(x-> abs(D(N,x)), 0, 1)[1]\nG(N) = (2/π)^2*log(2N+1)+1\nn = [1,2,3,6,10,20,30,60,100,200,300,600]\nplot(size=(500,350), legend=:topleft, xlabel=\"N\", xscale=:log)\nplot!(n, F.(n), label=\"integal of D_N(x)\", lw=2)\nplot!(n, G.(n), label=\"(2/pi)^2*log(2N+1)+1\", lw=2, ls=:dash)",
      "execution_count": 6,
      "outputs": [
        {
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stroke-width:1; stroke-opacity:1; fill:none\" points=\"\n  47.4745,72.051 208.784,72.051 208.784,26.691 47.4745,26.691 47.4745,72.051 \n  \"/>\n<polyline clip-path=\"url(#clip7100)\" style=\"stroke:#009af9; stroke-width:2; stroke-opacity:1; fill:none\" points=\"\n  52.4745,41.811 82.4745,41.811 \n  \"/>\n<g clip-path=\"url(#clip7100)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 87.4745, 46.311)\" x=\"87.4745\" y=\"46.311\">integal of D_N(x)</text>\n</g>\n<polyline clip-path=\"url(#clip7100)\" style=\"stroke:#e26f46; stroke-width:2; stroke-opacity:1; fill:none\" stroke-dasharray=\"8, 5\" points=\"\n  52.4745,56.931 82.4745,56.931 \n  \"/>\n<g clip-path=\"url(#clip7100)\">\n<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 87.4745, 61.431)\" x=\"87.4745\" y=\"61.431\">(2/pi)^2*log(2N+1)+1</text>\n</g>\n</svg>\n"
          },
          "execution_count": 6,
          "metadata": {},
          "output_type": "execute_result"
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      "cell_type": "markdown",
      "source": "### Fourier級数に関するRiemannの局所性定理\n\n**Riemannの局所性定理:** $f$ は $\\R$ 上の周期 $1$ を持つ函数であり, $[0,1)$ 上で可積分であると仮定し, $0<\\delta<1/2$ であるとする. このとき, \n\n$$\n\\lim_{N\\to\\infty}\\int_{-1/2}^{-\\delta} f(x+y)D_N(y)\\,dy = 0, \\quad\n\\lim_{N\\to\\infty}\\int_\\delta^{1/2} f(x+y)D_N(y)\\,dy = 0.\n$$\n\nこれより, $\\sum_{k=-N}^N a_n(f)e^{2\\pi ikx} = \\int_{-1/2}^{1/2} f(x+y) D_N(y)\\,dy$ が $N\\to\\infty$ で収束することと, $\\int_{-\\delta}^\\delta f(x+y) D_N(y)\\,dy$ が $N\\to\\infty$ で収束することは同値であり, 収束する場合には同じ値に収束することがわかる.  後者が収束するか否かは $x$ の近くでの函数 $f$ の様子だけで決まることに注意せよ. \n\n**証明:**\n$$\n\\begin{aligned}\n&\n\\int_{-1/2}^{-\\delta} f(x+y)D_N(y)\\,dy = \\int_{-1/2}^{-\\delta} \\frac{f(x+y)}{\\sin(\\pi y)}\\sin(\\pi(2N+1)y\\,dy,\n\\\\ &\n\\int_\\delta^{1/2} f(x+y)D_N(y)\\,dy = \\int_\\delta^{1/2} \\frac{f(x+y)}{\\sin(\\pi y)}\\sin(\\pi(2N+1)y\\,dy.\n\\end{aligned}\n$$\n\nであり, 区間 $(-1/2,-\\delta)$, $(\\delta,1/2)$ で $\\ds \\frac{f(x+y)}{\\sin(\\pi y)}$ は可積分なので, Riemann-Lebesgueの定理より, これらは $N\\to\\infty$ で $0$ に収束する. $\\QED$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### Fourier級数の収束 (Diniの条件)\n\n**定理(Diniの条件):** $f$ は $\\R$ 上の周期 $1$ を持つ函数であり, 区間 $[0,1)$ 上で可積分であると仮定し, $x\\in \\R$ であるとする. さらに, $0<\\delta<1/2$ を満たすある実数 $\\delta$ が存在して,  \n\n$$\n\\int_0^\\delta \\frac{|(f(x+y) + f(x-y) - 2f(x)|}{y}\\,dy <\\infty\n\\tag{$*$}\n$$\n\nが成立していると仮定する. この条件($*$)を**Diniの条件**と呼ぶ. このとき, \n\n$$\n\\lim_{N\\to\\infty}\\sum_{k=-N}^N a_n(f)e^{2\\pi ikx} = \\lim_{N\\to\\infty}\\int_{-1/2}^{1/2} f(x+y) D_N(y)\\,dy = f(x).\n$$\n\n**証明:** Fourier変換の場合と同様に計算すると, \n\n$$\n\\begin{aligned}\n\\sum_{k=-N}^N a_n(f)e^{2\\pi ikx} - f(x) &=\n\\int_{-1/2}^{1/2} f(x+y) D_N(y)\\,dy - \\int_{-1/2}^{1/2}D_N(y)\\,dy\n\\\\ &=\n\\int_{-1/2}^{1/2} (f(x+y)-f(x)) D_N(y)\\, dy\n\\\\ &=\n\\int_0^{1/2} (f(x+y)-f(x)) D_N(y)\\, dy +\n\\int_{-1/2}^0 (f(x+y)-f(x)) D_N(y)\\, dy\n\\\\ &=\n\\int_0^{1/2} (f(x+y)-f(x)) D_N(y)\\, dy +\n\\int_0^{1/2} (f(x-y)-f(x)) D_N(y)\\, dy\n\\\\ &=\n\\int_0^{1/2} \\frac{f(x+y)+f(x-y)-2f(x)}{y} \\frac{y}{\\sin(\\pi y)} \\sin(\\pi(2N+1)y)\\,dy.\n\\end{aligned}\n$$\n\nDiniの条件より, Riemann-Lebesgueの定理を使用できるので, これは $N\\to\\infty$ で $0$ に収束することがわかる. $\\QED$"
    },
    {
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      "cell_type": "markdown",
      "source": "## Poissonの和公式"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### Poissonの和公式とその証明\n\nFourier級数の収束とFourier変換の定義から, **Poissonの和公式**が得られることを説明しよう."
    },
    {
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      "cell_type": "markdown",
      "source": "**定義(急減少函数):** $\\R$ 上の $C^\\infty$ 函数 $f$ で, 任意の非負の整数 $m,n$ に対して, $|x|\\to\\infty$ のとき $f^{(m)}(x)x^n\\to 0$ となるものを**急減少函数**と呼ぶ. $\\QED$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "**定理(Poissonの和公式)** $f$ は $\\R$ 上の急減少函数であると仮定する. このとき, 次の公式が成立している:\n\n$$\n\\sum_{m\\in\\Z} f(m) = \\sum_{n\\in\\Z} \\hat{f}(n)\n$$\n\n**略証:** $\\ds g(x) = \\sum_{m\\in\\Z} f(x+m)$ とおくと, $g(x)$ は周期 $1$ を持つ $C^\\infty$ 函数になる. ゆえに,\n\n$$\ng(x) = \\sum_{n\\in\\Z} a_n(g)e^{2\\pi inx} \\quad (x\\in \\R).\n$$\n\nそして, \n\n$$\n\\begin{aligned}\na_n(g) &= \\int_0^1 \\sum_{m\\in\\Z}f(x+m)e^{-2\\pi inx}\\,dx =\n\\sum_{m\\in\\Z} \\int_0^1 f(x+m)e^{-2\\pi inx}\\,dx \n\\\\ &=\n\\sum_{m\\in\\Z} \\int_m^{m+1} f(x)e^{-2\\pi inx}\\,dx =\n\\int_{-\\infty}^\\infty f(x)e^{-2\\pi inx}\\,dx = \\hat{f}(n).\n\\end{aligned}\n$$\n\n以上をまとめると,\n\n$$\ng(x) = \\sum_{m\\in\\Z} f(x+m) = \\sum_{n\\in\\Z} \\hat{f}(n)e^{2\\pi inx} \\quad (x\\in\\R).\n$$\n\nこの等式で $x=0$ をおけば欲しい公式が得られる. $\\QED$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "### モジュラー変換性を持つ函数の例\n\n**問題:** $t>0$ に対して, 次の公式を示せ:\n\n$$\n\\sum_{m\\in\\Z} \\frac{e^{-\\pi(x-m)^2/t}}{\\sqrt{t}} = \\sum_{n\\in\\Z} e^{-\\pi n^2 t + 2\\pi inx}.\n$$\n\n**注意:** この問題の左辺と右辺は楕円テータ函数と呼ばれる特殊函数の特別な場合になっている. 左辺は熱方程式の基本解を周期函数になるように足し上げたものになっている. そのことから, この公式の両辺は周期境界条件のもとでの熱方程式の基本解になっていることがわかる. 楕円テータ函数については\n\n* David Mumford, <a href=\"http://www.dam.brown.edu/people/mumford/alg_geom/papers/Tata1.pdf\">Tata Lectures on Theta I</a>, Reprint of the 1983 Edition\n\nが非常に面白くてかつ読み易い教科書であり, 論文などでもよく引用されている. $\\QED$\n\n**解答例:** $\\ds f(x)=\\frac{e^{-\\pi x^2/t}}{\\sqrt{t}}$ とおくと, $f(x)$ が偶函数であることと, Poissonの和公式の証明より,\n\n$$\n\\sum_{m\\in\\Z} \\frac{e^{-\\pi(x-m)^2/t}}{\\sqrt{t}} =\n\\sum_{m\\in\\Z}f(x-m)=\\sum_{m\\in\\Z}f(x+m)=\\sum_{n\\in\\Z}\\hat{f}(n)e^{2\\pi inx}.\n$$\n\nそして, 公式\n\n$$\n\\int_{-\\infty}^\\infty e^{-y^2/a}e^{-ipy}\\,dy = \\sqrt{\\pi a}e^{-ap^2/4}\n$$\n\nより, $x=y/(2\\pi)$ とおくと, \n\n$$\n\\hat{f}(n) = \\int_{-\\infty}^\\infty \\frac{e^{-\\pi x^2/t}}{\\sqrt{t}} e^{-2\\pi inx}\\,dx =\n\\int_{-\\infty}^\\infty \\frac{e^{-y^2/(4\\pi t)}}{\\sqrt{t}} e^{-iny}\\,\\frac{dy}{2\\pi} =\n\\frac{\\sqrt{4\\pi t\\,\\pi}}{2\\pi\\sqrt{t}}e^{-4\\pi t n^2/4} =\ne^{-\\pi n^2 t}.\n$$\n\n以上をまとめると示したい公式が得られる. $\\QED$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "**問題(モジュラー変換性を持つ函数の例):** $t>0$ に対して, $\\ds \\Theta(t)=\\sum_{n\\in\\Z} e^{-\\pi n^2 t} = 1 + 2\\sum_{n=1}^\\infty e^{-\\pi n^2 t}$ とおくと, \n\n$$\n\\Theta(t) = \\frac{1}{\\sqrt{t}}\\Theta\\left(\\frac{1}{t}\\right)\n\\quad\\text{すなわち}\\quad\n\\Theta\\left(\\frac{1}{t}\\right) = \\sqrt{t}\\;\\Theta(t)\n$$\n\nが成立することを示せ.  この結果を $\\Theta(t)$ の**モジュラー変換性**と呼ぶ.\n\n**注意:** $\\Theta(t)$ の定義式を見ると, $t>0$ が大きなときに $\\Theta(t)$ がほぼ $1$ になることはすぐに確認できるが, $t>0$ が小さいときにはどのような様子になっているかはよくわからない.  しかし, モジュラー変換性を使うと, $\\Theta(t)$ は, $t>0$ が小さいときにはほぼ $\\ds\\frac{1}{\\sqrt{t}}$ に等しいことがわかる.  このようにモジュラー変換性は「よくわからない量」を「よくわかる量」に関係付ける公式である. $\\QED$\n\n**注意:** <a hred=\"https://www.google.co.jp/search?q=%E8%B6%85%E5%BC%A6%E7%90%86%E8%AB%96+%E3%83%A2%E3%82%B8%E3%83%A5%E3%83%A9%E3%83%BC%E5%A4%89%E6%8F%9B%E6%80%A7\">超弦理論</a>や<a href=\"https://www.google.co.jp/search?q=%E5%85%B1%E5%BD%A2%E5%A0%B4%E7%90%86%E8%AB%96+%E3%83%A2%E3%82%B8%E3%83%A5%E3%83%A9%E3%83%BC%E5%A4%89%E6%8F%9B%E6%80%A7\">共形場理論</a>ではモジュラー変換性が重要な意味を持つ. $\\QED$\n\n**注意:** 上の問題の公式はRiemannのゼータ函数の函数等式を証明するときに使われる. 上の問題の結果は楕円テータ函数のモジュラー変換性の特別な場合になっている. モジュラー変換性を持つ函数は数論的にも極めて重要な数学的対象である. $\\QED$\n\n**解答例1:**  $\\ds f(x)=\\frac{e^{-\\pi x^2/t}}{\\sqrt{t}}$ にPoissonの和公式を適用すればこの公式が得られる. すなわち, 上の問題の結果で $x=0$ とおけば示したい公式が得られる. $\\QED$\n\n**解答例2:** $f(x)=e^{-\\pi n^2 t}$ について $\\ds\\hat{f}(p) = \\frac{e^{-\\pi p^2/t}}{\\sqrt{t}}$ なので, Poissonの和公式より, 示したい公式が得られる. $\\QED$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "**問題:** 上の問題の結論を $\\Theta(t)$ と $\\ds\\frac{1}{\\sqrt{t}}\\Theta\\left(\\frac{1}{t}\\right)$ のグラフを重ねてプロットすることによって確認せよ. 2つのグラフはぴったり重なるはずである. $\\QED$\n\n次のセルを見よ."
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "Theta(t; N=20) = iszero(N) ? one(t) : 1 + 2*sum(n->e^(-π*n^2*t), 1:N)\nt = 0.2:0.05:2.0\nA = Theta.(t)\nB = 1./sqrt.(t).*Theta.(1./t)\nplot(size=(500, 350))\nplot!(t, B, label=\"Theta(t)\",               lw=2)\nplot!(t, A, label=\"(1/sqrt(t)) Theta(1/t)\", lw=2, ls=:dash)",
      "execution_count": 7,
      "outputs": [
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      "source": "**問題:** $t=0.1$ のとき, まず $\\ds\\frac{1}{\\sqrt{t}}\\Theta\\left(\\frac{1}{t}\\right)$ の近似値を\n\n$$\n\\frac{1}{\\sqrt{t}}\\left(1 + 2\\sum_{n=1}^N e^{-\\pi n^2/t}\\right)\n$$\n\nの $N=1$ の場合を用いて計算せよ. そして, $\\Theta(t)$ を\n\n$$\n1 + 2\\sum_{n=1}^N e^{-\\pi n^2 t}\n$$\n\nで近似するとき, $N\\geqq 10$ としなければ, 上と同じ精度が得られないことを確認せよ. $t>0$ が小さいときには $\\Theta(t)$ の定義に基いて数値計算するより, モジュラー変換を用いて数値計算した方が効率的である. $\\QED$\n\n次のセルを見よ."
    },
    {
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      "cell_type": "code",
      "source": "Theta(t; N=20) = iszero(N) ? one(t) : 1 + 2*sum(n->e^(-π*n^2*t), 1:N)\nt = 0.1\n@show 1/√t*Theta(1/t; N=0)\n@show 1/√t*Theta(1/t; N=1)\n@show 1/√t*Theta(1/t; N=2)\n@show Theta(t; N=9);\n@show Theta(t; N=10);\n@show Theta(t; N=11);",
      "execution_count": 8,
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": "(1 / √t) * Theta(1 / t; N=0) = 3.162277660168379\n(1 / √t) * Theta(1 / t; N=1) = 3.162277660168523\n(1 / √t) * Theta(1 / t; N=2) = 3.162277660168523\nTheta(t; N=9) = 3.1622776601684772\nTheta(t; N=10) = 3.1622776601685225\nTheta(t; N=11) = 3.1622776601685225\n"
        }
      ]
    },
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      },
      "cell_type": "code",
      "source": "",
      "execution_count": null,
      "outputs": []
    }
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