[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki3\/archives\/7790#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki3\/archives\/7790","headline":"\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u53ce\u675f &#8211; Wikipedia","name":"\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u53ce\u675f &#8211; Wikipedia","description":"\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u53ce\u675f\u306f\u7d14\u7c8b\u6570\u5b66\u306b\u304a\u3051\u308b\u8abf\u548c\u89e3\u6790\u306e\u5206\u91ce\u3067\u7814\u7a76\u3055\u308c\u308b\u554f\u984c\u3067\u3042\u308b\u3002\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306f\u4e00\u822c\u306b\u306f\u53ce\u675f\u3059\u308b\u3068\u306f\u9650\u3089\u305a\u3001\u53ce\u675f\u3059\u308b\u305f\u3081\u306e\u6761\u4ef6\u304c\u5b58\u5728\u3059\u308b\u3002 \u53ce\u675f\u6027\u306e\u5224\u65ad\u306b\u306f\u5404\u70b9\u53ce\u675f\u3001\u4e00\u69d8\u53ce\u675f\u3001\u7d76\u5bfe\u53ce\u675f\u3001L\u2009p \u7a7a\u9593\u3001\u7dcf\u548c\u6cd5\u3001\u30c1\u30a7\u30b6\u30ed\u548c\u306e\u77e5\u8b58\u3092\u8981\u3059\u308b\u3002 \u533a\u9593 [0, 2\u03c0] \u3067\u53ef\u7a4d\u5206\u306a f \u3092\u8003\u3048\u308b\u3002f \u306e\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570 (Fourier coefficient) f^(n){displaystyle {widehat {f}}(n)} \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u3081\u3089\u308c\u308b\u3002 f^(n)=12\u03c0\u222b02\u03c0f(t)e\u2212intdt,n\u2208Z.{displaystyle {widehat {f}}(n)={frac {1}{2pi","datePublished":"2020-12-19","dateModified":"2020-12-19","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki3\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki3\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/dcbd99c1d4c40b9ec9b6ff640ee2d48f16a92e07","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/dcbd99c1d4c40b9ec9b6ff640ee2d48f16a92e07","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/jp\/wiki3\/archives\/7790","about":["Wiki"],"wordCount":7259,"articleBody":"\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u53ce\u675f\u306f\u7d14\u7c8b\u6570\u5b66\u306b\u304a\u3051\u308b\u8abf\u548c\u89e3\u6790\u306e\u5206\u91ce\u3067\u7814\u7a76\u3055\u308c\u308b\u554f\u984c\u3067\u3042\u308b\u3002\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306f\u4e00\u822c\u306b\u306f\u53ce\u675f\u3059\u308b\u3068\u306f\u9650\u3089\u305a\u3001\u53ce\u675f\u3059\u308b\u305f\u3081\u306e\u6761\u4ef6\u304c\u5b58\u5728\u3059\u308b\u3002  \u53ce\u675f\u6027\u306e\u5224\u65ad\u306b\u306f\u5404\u70b9\u53ce\u675f\u3001\u4e00\u69d8\u53ce\u675f\u3001\u7d76\u5bfe\u53ce\u675f\u3001L\u2009p \u7a7a\u9593\u3001\u7dcf\u548c\u6cd5\u3001\u30c1\u30a7\u30b6\u30ed\u548c\u306e\u77e5\u8b58\u3092\u8981\u3059\u308b\u3002\u533a\u9593 [0, 2\u03c0] \u3067\u53ef\u7a4d\u5206\u306a f \u3092\u8003\u3048\u308b\u3002f \u306e\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570 (Fourier coefficient) f^(n){displaystyle {widehat {f}}(n)} \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u3081\u3089\u308c\u308b\u3002  f^(n)=12\u03c0\u222b02\u03c0f(t)e\u2212intdt,n\u2208Z.{displaystyle {widehat {f}}(n)={frac {1}{2pi }}int _{0}^{2pi }f(t)e^{-int},dt,quad nin mathbf {Z} .}\u95a2\u6570 f \u3068\u305d\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u95a2\u4fc2\u306f\u901a\u5e38\u6b21\u306e\u3088\u3046\u306b\u8a18\u8ff0\u3055\u308c\u308b\u3002f\u223c\u2211nf^(n)eint.{displaystyle fsim sum _{n}{widehat {f}}(n)e^{int}.}\u3053\u3053\u3067 \u223c \u306f\u548c\u304c\u3042\u308b\u610f\u5473\u3067\u95a2\u6570\u3092\u8868\u73fe\u3059\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002\u3088\u308a\u614e\u91cd\u306a\u8b70\u8ad6\u3092\u8981\u3059\u308b\u5834\u5408\u306b\u306f\u3001\u90e8\u5206\u548c\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b\uff1a  SN(f;t)=\u2211n=\u2212NNf^(n)eint.{displaystyle S_{N}(f;t)=sum _{n=-N}^{N}{widehat {f}}(n)e^{int}.}\u3053\u306e\u3068\u304d\u6c17\u306b\u306a\u308b\u3067\u3042\u308d\u3046\u554f\u984c\u306f\u6b21\u306e\u4e8b\u3067\u3042\u308b\uff1a\u95a2\u6570 SN(f;t) \u306f f \u3078\u3001\u307e\u305f\u3069\u306e\u610f\u5473\u3067\u53ce\u675f\u3059\u308b\u3060\u308d\u3046\u304b\uff1f\u3000\u53ce\u675f\u3092\u4fdd\u8a3c\u3059\u308b f \u306e\u6761\u4ef6\u306f\u4f55\u3060\u308d\u3046\u304b\uff1f\u3053\u306e\u8a18\u4e8b\u3067\u306f\u3053\u308c\u3089\u306e\u554f\u306b\u95a2\u3059\u308b\u8b70\u8ad6\u3092\u4e3b\u3068\u3057\u3066\u6271\u3046\u3002\u5148\u3092\u7d9a\u3051\u308b\u524d\u306b\u30c7\u30a3\u30ea\u30af\u30ec\u6838 (Dirichlet kernel) \u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u3066\u304a\u304f\u3002\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570 f^(n){displaystyle {widehat {f}}(n)} \u306e\u516c\u5f0f\u3092\u90e8\u5206\u548c SN \u306b\u5bfe\u3057\u3066\u9069\u7528\u3059\u308b\u3068\u3001\u6700\u7d42\u7684\u306bSN(f)=f\u2217DN{displaystyle S_{N}(f)=f*D_{N},}\u3068\u3044\u3046\u95a2\u4fc2\u304c\u5f97\u3089\u308c\u308b\u3002\u3053\u3053\u3067 \u2217 \u306f\u5de1\u56de\u7573\u307f\u8fbc\u307f\u3092\u610f\u5473\u3057\u3001DN \u306f\u4ee5\u4e0b\u306b\u793a\u3059\u30c7\u30a3\u30ea\u30af\u30ec\u6838\u3067\u3042\u308b\uff1aDn(t)=sin\u2061((n+12)t)sin\u2061(t\/2).{displaystyle D_{n}(t)={frac {sin((n+{frac {1}{2}})t)}{sin(t\/2)}}.}\u30c7\u30a3\u30ea\u30af\u30ec\u6838\u306f\u6b63\u5024\u3067\u306f\u306a\u304f \u3001\u5b9f\u969b\u305d\u306e\u30ce\u30eb\u30e0\u306f\u767a\u6563\u3059\u308b\u3002\u222b|Dn(t)|dt\u2192\u221e{displaystyle int |D_{n}(t)|,dtto infty }\u3053\u306e\u6027\u8cea\u306f\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u53ce\u675f\u306b\u95a2\u3059\u308b\u8b70\u8ad6\u3067\u6975\u3081\u3066\u91cd\u8981\u306a\u5f79\u5272\u3092\u679c\u305f\u3059\u3002L\u20091(T) \u4e0a\u306e Dn \u306e\u30ce\u30eb\u30e0\u306f\u3001C(T) \u7a7a\u9593\u306e\u5468\u671f\u7684\u9023\u7d9a\u95a2\u6570\u306b\u4f5c\u7528\u3059\u308b Dn \u7573\u307f\u8fbc\u307f\u4f5c\u7528\u7d20\u306e\u30ce\u30eb\u30e0\u3068\u4e00\u81f4\u3057\u3001\u307e\u305f C(T) \u4e0a\u306e\u7dda\u578b\u6c4e\u95a2\u6570 \u0192\u00a0\u2192 (Sn\u0192)(0) \u306e\u30ce\u30eb\u30e0\u306b\u4e00\u81f4\u3059\u308b\u3002\u5f93\u3063\u3066\u3001\u3053\u306e C(T) \u4e0a\u306e\u7dda\u578b\u6c4e\u95a2\u6570\u306e\u65cf\u306f n\u00a0\u2192\u00a0\u221e \u3068\u3057\u305f\u3068\u304d\u306b\u53ce\u675f\u3057\u306a\u3044\u3002Table of Contents\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u306e\u5927\u304d\u3055[\u7de8\u96c6]\u5404\u70b9\u53ce\u675f\u3059\u308b\u305f\u3081\u306e\u6761\u4ef6[\u7de8\u96c6]\u305d\u306e\u70b9\u3067\u5de6\u5fae\u5206\u3068\u53f3\u5fae\u5206\u3092\u6301\u3064\u5834\u5408[\u7de8\u96c6]\u30d8\u30eb\u30c0\u30fc\u6761\u4ef6[\u7de8\u96c6]\u305d\u306e\u4ed6[\u7de8\u96c6]\u4e00\u69d8\u53ce\u675f\u3059\u308b\u305f\u3081\u306e\u6761\u4ef6[\u7de8\u96c6]\u7d76\u5bfe\u53ce\u675f\u3059\u308b\u305f\u3081\u306e\u6761\u4ef6[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u6559\u79d1\u66f8[\u7de8\u96c6]\u8ad6\u6587[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u306e\u5927\u304d\u3055[\u7de8\u96c6]\u5fdc\u7528\u306b\u304a\u3044\u3066\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u306e\u5927\u304d\u3055\u3092\u77e5\u308b\u3053\u3068\u304c\u3057\u3070\u3057\u3070\u91cd\u8981\u306b\u306a\u308b\u3002\u95a2\u6570 f \u304c\u7d76\u5bfe\u9023\u7d9a\u3067\u3042\u308b\u306a\u3089\u3001\u95a2\u6570 f \u306e\u307f\u306b\u4f9d\u5b58\u3059\u308b\u5b9a\u6570 K \u306b\u3064\u3044\u3066\u3001\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3064\u3002|f^(n)|\u2264K|n|.{displaystyle left|{widehat {f}}(n)right|leq {K over |n|}.}f \u304c\u6709\u754c\u5909\u52d5\u95a2\u6570\uff08\u82f1\u8a9e\u7248\uff09\u3067\u3042\u308b\u306a\u3089\u3001\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3064\u3002|f^(n)|\u2264var(f)2\u03c0|n|.{displaystyle left|{widehat {f}}(n)right|leq {{rm {var}}(f) over 2pi |n|}.}f \u2208 C\u2009p \u306a\u3089\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3064\u3002|f^(n)|\u2264\u2016f(p)\u2016L1|n|p.{displaystyle left|{widehat {f}}(n)right|leq {|f^{(p)}|_{L_{1}} over |n|^{p}}.}f \u2208 C\u2009p \u304b\u3064 f\u2009(p) \u304c \u03c9p \u306e\u9023\u7d9a\u7387\uff08\u82f1\u8a9e\u7248\uff09\u3092\u6301\u3064\u306a\u3089[\u8981\u51fa\u5178]\u3001|f^(n)|\u2264\u03c9(2\u03c0\/n)|n|p{displaystyle left|{widehat {f}}(n)right|leq {omega (2pi \/n) over |n|^{p}}}\u304c\u6210\u308a\u7acb\u3064\u3002\u5f93\u3063\u3066\u3001f \u306f \u03b1&#8211;\u30d8\u30eb\u30c0\u30fc\u30af\u30e9\u30b9\uff08\u82f1\u8a9e\u7248\uff09\u3067\u3042\u308b\uff08\u30ea\u30d7\u30b7\u30c3\u30c4\u9023\u7d9a\u3082\u53c2\u7167\uff09\u3002|f^(n)|\u2264K|n|\u03b1.{displaystyle left|{widehat {f}}(n)right|leq {K over |n|^{alpha }}.}\u5404\u70b9\u53ce\u675f\u3059\u308b\u305f\u3081\u306e\u6761\u4ef6[\u7de8\u96c6]  \u6b63\u5f26\u6ce2\uff08\u4e0b\uff09\u3092\u57fa\u5e95\u3068\u3057\u305f\u91cd\u306d\u5408\u308f\u305b\u306b\u3088\u3063\u3066\u4f5c\u3089\u308c\u305f\u30ce\u30b3\u30ae\u30ea\u6ce2\uff08\u4e0a\uff09\uff1b\u57fa\u5e95\u3068\u306a\u308b\u6b63\u5f26\u6ce2\u306e\u6ce2\u9577 \u03bb\/k \u306f\u30ce\u30b3\u30ae\u30ea\u6ce2\u306e\u6ce2\u9577 \u03bb \u3088\u308a\u77ed\u3044\uff08k \u306f 1 \u3088\u308a\u5927\u304d\u3044\u6574\u6570\uff09\u3002\u3059\u3079\u3066\u306e\u57fa\u5e95\u306f\u30ce\u30b3\u30ae\u30ea\u6ce2\u3068\u540c\u3058\u70b9\u306b\u7bc0\u3092\u6301\u3064\u304c\u3001\u539f\u7406\u7684\u306b\u3059\u3079\u3066\u306e\u57fa\u5e95\u306f\u4f59\u8a08\u306b\u7bc0\u3092\u3064\u304f\u3063\u3066\u3057\u307e\u3046\u3002\u30ce\u30b3\u30ae\u30ea\u6ce2\u306e\u632f\u52d5\u73fe\u8c61\u306f\u30ae\u30d6\u30ba\u73fe\u8c61\u3068\u547c\u3070\u308c\u3066\u3044\u308b\u3002\u305d\u306e\u70b9\u3067\u5de6\u5fae\u5206\u3068\u53f3\u5fae\u5206\u3092\u6301\u3064\u5834\u5408[\u7de8\u96c6]\u70b9 x_0 \u3092\u4e0e\u3048\u305f\u3068\u304d\u3001\u305d\u306e\u70b9\u3067\u95a2\u6570\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u304c\u53ce\u675f\u3059\u308b\u5341\u5206\u6761\u4ef6\u306b\u3064\u3044\u3066\u306f\u6b21\u304c\u3088\u304f\u77e5\u3089\u308c\u3066\u3044\u308b\uff1bf \u304c\u5468\u671f 2\u03c0 \u306e\u533a\u5206\u7684\u306b C1 \u7d1a\u306e\u53ef\u7a4d\u5206\u95a2\u6570\u3067\u3042\u308a\u3001\u70b9x_0\u3067\u306e\u5de6\u5fae\u5206\u3068\u53f3\u5fae\u5206\u3092\u6301\u3064\u3068\u3059\u308b\u3002\u3053\u306e\u3068\u304df\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306f12(f(x0\u22120)+f(x0+0)){displaystyle {frac {1}{2}}left(f(x_{0}-0)+f(x_{0}+0)right)}\u306b\u53ce\u675f\u3059\u308b\uff08\u3053\u3053\u3067f (x \u00b1 0) = limh \u2193 0f (x \u00b1 h) \uff09\u3002\u3064\u307e\u308a\u305f\u3068\u3048\u8df3\u8e8d\u4e0d\u9023\u7d9a\u70b9\u3067\u3042\u3063\u3066\u3082\u3001\u95a2\u6570\u304c\u305d\u3053\u3067\u5de6\u5fae\u5206\u3068\u53f3\u5fae\u5206\u3092\u6301\u3064\u5834\u5408\u3001\u305d\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306f\u305d\u3053\u3067\u306e\u5de6\u6975\u9650\u5024\u3068\u53f3\u6975\u9650\u5024\u306e\u3061\u3087\u3046\u3069\u4e2d\u9593\u306b\u53ce\u675f\u3059\u308b\uff08\u30ae\u30d6\u30ba\u73fe\u8c61\u3082\u53c2\u7167\uff09\u3002\u30d8\u30eb\u30c0\u30fc\u6761\u4ef6[\u7de8\u96c6]\u30c7\u30a3\u30ea\u30af\u30ec\uff1d\u30c7\u30a3\u30cb\u6761\u4ef6 (Dirichlet\u2013Dini criterion)f \u304c 2\u03c0-\u5468\u671f\u7684\u3067\u3042\u308a\u3001\u5c40\u6240\u53ef\u7a4d\u5206\u304b\u3064\u6b21\u306e\u6761\u4ef6\u222b0\u03c0|f(x0+t)+f(x0\u2212t)2\u2212\u2113|dtt(SNf)(x)|\u2264Kln\u2061NNp\u03c9(2\u03c0\/N).{displaystyle |f(x)-(S_{N}f)(x)|leq K{ln N over N^{p}}omega (2pi \/N).}\u3053\u3053\u3067 K \u306f f \u306b\u3082 p \u306b\u3082 N \u306b\u3082\u4f9d\u5b58\u3057\u306a\u3044\u5b9a\u6570\u3067\u3042\u308b\u3002\u3053\u306e\u5b9a\u7406\u306f\u3001\u4f8b\u3048\u3070 f \u304c \u03b1-\u30d8\u30eb\u30c0\u30fc\u6761\u4ef6\u3092\u6e80\u305f\u3059\u5834\u5408\u3001|f(x)\u2212(SNf)(x)|\u2264Kln\u2061NN\u03b1{displaystyle |f(x)-(S_{N}f)(x)|leq K{ln N over N^{alpha }}}\u3067\u62bc\u3055\u3048\u3089\u308c\u308b\u3053\u3068\u3092\u793a\u3059\u3002f \u304c 2\u03c0 \u5468\u671f\u7684\u3067\u3042\u308a [0, 2\u03c0] \u3067\u7d76\u5bfe\u9023\u7d9a\u306a\u3089\u3070\u3001\u95a2\u6570 f \u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306f f \u306b\u4e00\u69d8\u53ce\u675f\u3059\u308b\u3002\u305f\u3060\u3057\u7d76\u5bfe\u53ce\u675f\u3059\u308b\u3068\u306f\u9650\u3089\u306a\u3044[3]\u3002\u7d76\u5bfe\u53ce\u675f\u3059\u308b\u305f\u3081\u306e\u6761\u4ef6[\u7de8\u96c6]\u95a2\u6570 f \u304c\u7d76\u5bfe\u53ce\u675f\u3059\u308b\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u3092\u6301\u3064\u5834\u5408\u3001\u2016f\u2016A:=\u2211n=\u2212\u221e\u221e|f^(n)|f\u2016A\u2264c\u03b1\u2016f\u2016Lip\u03b1,{displaystyle |f|_{A}leq c_{alpha }|f|_{{rm {Lip}}_{alpha }},}\u2016f\u2016K:=\u2211n=\u2212\u221e+\u221e|n||f^(n)|2\u2264c\u03b1\u2016f\u2016Lip\u03b12,{displaystyle |f|_{K}:=sum _{n=-infty }^{+infty }|n||{widehat {f}}(n)|^{2}leq c_{alpha }|f|_{{rm {Lip}}_{alpha }}^{2},}\u304c\u6210\u308a\u7acb\u3064\u3002\u307e\u305f ||f\u2009||K \u306f\u30af\u30ec\u30a4\u30f3\u4ee3\u6570\u306b\u304a\u3051\u308b\u30ce\u30eb\u30e0\u3067\u3042\u308b\u3002\u6761\u4ef6\u306b\u3042\u3063\u305f 1\/2 \u304c\u57fa\u672c\u7684\u306a\u5f79\u5272\u3092\u679c\u305f\u3057\u3066\u3044\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u30021\/2 \u30d8\u30eb\u30c0\u30fc\u95a2\u6570\u306f\u30a6\u30a3\u30fc\u30ca\u30fc\u4ee3\u6570\u306b\u5c5e\u3055\u306a\u3044\u306e\u3067\u3042\u308b\u3002\u307e\u305f\u3053\u306e\u5b9a\u7406\u306f\u3001\u3088\u304f\u77e5\u3089\u308c\u3066\u3044\u308b \u03b1-\u30d8\u30eb\u30c0\u30fc\u95a2\u6570\u306e\u30d5\u30fc\u30ea\u30a8\u4fc2\u6570\u306e\u5927\u304d\u3055\u306e\u4e0a\u9650\u3001O(1\/n\u03b1) \u3092\u6539\u826f\u3059\u308b\u3053\u3068\u306f\u3067\u304d\u305a\u3001\u3053\u306e\u3068\u304d\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306f\u7dcf\u548c\u53ef\u80fd\u3067\u306f\u306a\u3044\u3002f \u304c\u6709\u754c\u5909\u52d5\u95a2\u6570\u3067\u3042\u308a\u304b\u3064\u3042\u308b \u03b1 > 0 \u306b\u3064\u3044\u3066 \u03b1-\u30d8\u30eb\u30c0\u30fc\u30af\u30e9\u30b9\u306b\u5c5e\u3059\u308b\u306a\u3089\u3001\u95a2\u6570 f \u306f\u30a6\u30a3\u30fc\u30ca\u30fc\u4ee3\u6570\u306b\u5c5e\u3059\u308b\u3002^ Antoni Zygmund, Trigonometric Series, vol. 1, Chapter 8, Theorem 1.13, p.\u00a0300 \u53c2\u7167\u3002^ Jackson (1930), p21ff.^ Stromberg (1981), Exercise 6 (d) on p.\u00a0519 and Exercise 7 (c) on p.\u00a0520.\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u6559\u79d1\u66f8[\u7de8\u96c6]Dunham Jackson (1930), The theory of Approximation, AMS Colloquium Publication Volume XI, New York\u00a0.Nina K. Bary (1964), A treatise on trigonometric series, I, II, Pergamon Press\u00a0. Authorized translation by Margaret F. Mullins.Antoni Zygmund (2002), Trigonometric series, I, II (Third ed.), Cambridge University Press, Cambridge, ISBN\u00a00-521-89053-5\u00a0 With a foreword by Robert A. Fefferman. Cambridge Mathematical Library.Yitzhak Katznelson, An introduction to harmonic analysis, Third edition. Cambridge University Press, Cambridge, 2004. ISBN 0-521-54359-2Karl R. Stromberg, &#8220;Introduction to classical analysis&#8221;, Wadsworth International Group, 1981. ISBN 0-534-98012-0The Katznelson book is the one using the most modern terminology and style of the three. The original publishing dates are: Zygmund in 1935, Bari in 1961 and Katznelson in 1968. Zygmund&#8217;s book was greatly expanded in its second publishing in 1959, however.\u8ad6\u6587[\u7de8\u96c6]Paul du Bois-Reymond, Ueber die Fourierschen Reihen, Nachr. K\u00f6n. Ges. Wiss. G\u00f6ttingen 21 (1873), 571\u2013582.This is the first proof that the Fourier series of a continuous function might diverge. In GermanAndrey Kolmogorov, Une s\u00e9rie de Fourier\u2013Lebesgue divergente presque partout, Fundamenta math. 4 (1923), 324\u2013328.Andrey Kolmogorov, Une s\u00e9rie de Fourier\u2013Lebesgue divergente partout, C. R. Acad. Sci. Paris 183 (1926), 1327\u20131328The first is a construction of an integrable function whose Fourier series diverges almost everywhere. The second is a strengthening to divergence everywhere. In French.Lennart Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966) 135\u2013157.Richard A. Hunt, On the convergence of Fourier series, Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967), 235\u2013255. Southern Illinois Univ. Press, Carbondale, Ill.Charles Louis Fefferman, Pointwise convergence of Fourier series, Ann. of Math. 98 (1973), 551\u2013571.Michael Lacey and Christoph Thiele, A proof of boundedness of the Carleson operator, Math. Res. Lett. 7:4 (2000), 361\u2013370.Ole G. J\u00f8rsboe and Leif Mejlbro, The Carleson\u2013Hunt theorem on Fourier series. Lecture Notes in Mathematics 911, Springer-Verlag, Berlin-New York, 1982. ISBN 3-540-11198-0This is the original paper of Carleson, where he proves that the Fourier expansion of any continuous function converges almost everywhere; the paper of Hunt where he generalizes it to Lp{displaystyle L^{p}} spaces; two attempts at simplifying the proof; and a book that gives a self contained exposition of it.Dunham Jackson, Fourier Series and Orthogonal Polynomials, 1963D. J. Newman,  A simple proof of Wiener&#8217;s 1\/f theorem,  Proc. Amer. Math. Soc.  48  (1975), 264\u2013265.Jean-Pierre Kahane and Yitzhak Katznelson, Sur les ensembles de divergence des s\u00e9ries trigonom\u00e9triques, Studia Math. 26 (1966), 305\u2013306In this paper the authors show that for any set of zero measure there exists a continuous function on the circle whose Fourier series diverges on that set. In French.Sergei Vladimirovich Konyagin, On divergence of trigonometric Fourier series everywhere, C. R. Acad. Sci. Paris 329 (1999), 693\u2013697.Jean-Pierre Kahane, Some random series of functions, second edition. Cambridge University Press, 1993. ISBN 0-521-45602-9The Konyagin paper proves the log\u2061n{displaystyle {sqrt {log n}}} divergence result discussed above. A simpler proof that gives only log\u00a0log\u00a0n can be found in Kahane&#8217;s book.\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki3\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki3\/archives\/7790#breadcrumbitem","name":"\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306e\u53ce\u675f &#8211; Wikipedia"}}]}]