{"id":13992,"date":"2024-04-28T23:22:16","date_gmt":"2024-04-28T23:22:16","guid":{"rendered":"https:\/\/science-hub.click\/%E3%83%9D%E3%82%A2%E3%82%BD%E3%83%B3%E5%92%8C%E3%81%AE%E5%85%AC%E5%BC%8F-%E5%AE%9A%E7%BE%A9\/"},"modified":"2024-04-28T23:22:16","modified_gmt":"2024-04-28T23:22:16","slug":"%E3%83%9D%E3%82%A2%E3%82%BD%E3%83%B3%E5%92%8C%E3%81%AE%E5%85%AC%E5%BC%8F-%E5%AE%9A%E7%BE%A9","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=13992","title":{"rendered":"\u30dd\u30a2\u30bd\u30f3\u548c\u306e\u516c\u5f0f – \u5b9a\u7fa9"},"content":{"rendered":"

\u5c0e\u5165<\/h2>

\u30dd\u30a2\u30bd\u30f3\u7dcf\u548c\u306e\u516c\u5f0f<\/b>(\u30dd\u30a2\u30bd\u30f3 \u30ec\u30b5\u30e0<\/b>\u3068\u547c\u3070\u308c\u308b\u3053\u3068\u3082\u3042\u308a\u307e\u3059) \u306f 2 \u3064\u306e\u7121\u9650\u548c\u306e\u9593\u306e\u6052\u7b49\u5f0f\u3067\u3042\u308a\u30011 \u3064\u76ee\u306f\u95a2\u6570f<\/i><\/span>\u3067\u69cb\u7bc9\u3055\u308c\u30012 \u3064\u76ee\u306f\u305d\u306e\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3067\u69cb\u7bc9\u3055\u308c\u307e\u3059\u3002

$$ {\\hat f} $$<\/div> \u3002\u3053\u3053\u3067f \u306f<\/i><\/span>\u5b9f\u8ef8\u4e0a\u306e\u95a2\u6570\u3001\u3088\u308a\u4e00\u822c\u7684\u306b\u306fn<\/i><\/span>\u6b21\u5143<\/a><\/span>\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593<\/a><\/span>\u4e0a\u306e\u95a2\u6570\u3067\u3059\u3002\u3053\u306e\u516c\u5f0f\u306fSim\u00e9on Denis Poisson<\/a><\/span>\u306b\u3088\u3063\u3066\u767a\u898b\u3055\u308c\u307e\u3057\u305f\u3002<\/p>

\u3053\u308c\u3068\u305d\u306e\u4e00\u822c\u5316\u306f\u3001\u6570\u8ad6<\/a><\/span>\u3001\u8abf\u548c<\/a><\/span>\u89e3\u6790\u3001\u30ea\u30fc\u30de\u30f3\u5e7e\u4f55\u5b66<\/a><\/span>\u306a\u3069\u306e\u6570\u5b66<\/a><\/span>\u306e\u3044\u304f\u3064\u304b\u306e\u5206\u91ce\u3067\u91cd\u8981\u3067\u3059\u3002 1 \u6b21\u5143\u306e\u5f0f\u3092\u89e3\u91c8\u3059\u308b 1 \u3064\u306e\u65b9\u6cd5\u306f\u3001\u5186<\/a><\/span>\u4e0a\u306e\u30e9\u30d7\u30e9\u30b9 \u30d9\u30eb\u30c8\u30e9\u30df\u6f14\u7b97\u5b50<\/a><\/span>\u306e\u30b9\u30da\u30af\u30c8\u30eb\u3068\u3053\u306e\u66f2\u7dda<\/a><\/span>\u4e0a\u306e\u5468\u671f\u6e2c\u5730\u7dda\u306e\u9577\u3055\u3068\u306e\u9593\u306e\u95a2\u4fc2\u3068\u3057\u3066\u898b\u308b\u3053\u3068\u3067\u3059\u3002\u30bb\u30eb\u30d0\u30fc\u30b0\u306e\u30c8\u30ec\u30fc\u30b9<\/a><\/span>\u516c\u5f0f\u306f\u3001\u4e0a\u3067\u5f15\u7528\u3057\u305f\u3059\u3079\u3066\u306e\u5206\u91ce\u3068\u95a2\u6570<\/a><\/span>\u89e3\u6790\u306e\u63a5\u70b9<\/a><\/span>\u3067\u3001\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3 \u30b9\u30da\u30af\u30c8\u30eb\u3068\u5b9a\u6570\u3092\u3082\u3064\u8868\u9762\u4e0a\u306e\u6e2c\u5730\u7dda\u306e\u9577\u3055\u3068\u306e\u9593\u306b\u3001\u540c\u3058\u30bf\u30a4\u30d7\u306e\u3001\u3057\u304b\u3057\u3088\u308a\u6df1\u3044\u7279\u5fb4\u3092\u6301\u3064\u95a2\u4fc2\u3092\u78ba\u7acb\u3057\u307e\u3059\u3002\u8ca0\u306e\u66f2\u7387<\/a><\/span>( n<\/i><\/span>\u6b21\u5143\u306e\u30dd\u30a2\u30bd\u30f3<\/a><\/span>\u516c\u5f0f\u306f\u3001\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u3068\u3001\u66f2\u7387\u30bc\u30ed\u306e\u7a7a\u9593\u3067\u3042\u308b\u30c8\u30fc\u30e9\u30b9\u306e\u5468\u671f\u6e2c\u5730\u7dda\u306b\u30ea\u30f3\u30af\u3055\u308c\u3066\u3044\u307e\u3059)\u3002<\/p>

\n\"\u30dd\u30a2\u30bd\u30f3\u548c\u306e\u516c\u5f0f<\/figure>

\u30dd\u30a2\u30bd\u30f3\u548c\u306e\u516c\u5f0f<\/h2>

\u8a55\u4fa1<\/span><\/h3>

\u307e\u305f\u306f\u95a2\u6570

$$ {f\\,} $$<\/div>\u305d\u306e\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059
$$ {\\hat{f}} $$<\/div> \u3001\u3064\u307e\u308a: <\/p>
$$ { f(x) = {1\\over 2\\pi}\\int_{-\\infty}^{\\infty} \\hat{f}(\\omega) \\ e^{ i x \\omega} \\ d \\omega } $$<\/div> \u3001<\/center>
\u305d\u3057\u3066
$$ { \\hat{f}(\\omega) = \\int_{-\\infty}^{\\infty} f(x) \\ e^{- i x \\omega} \\ dx } $$<\/div> \u3002<\/center>
\n\"\u30dd\u30a2\u30bd\u30f3\u548c\u306e\u516c\u5f0f<\/figure>

\u5b9a\u7406<\/span><\/span><\/h3>

f \u3092<\/i>\u8907\u7d20\u95a2\u6570\u3068\u3057\u307e\u3059\u3002

$$ {\\R} $$<\/div> \uff12\u56de\u9023\u7d9a\u5fae\u5206\u53ef\u80fd\u3002 f<\/i>\u3068\u305d\u306e\u6700\u521d\u306e 2 \u3064\u306e\u5c0e\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u4eee\u5b9a\u3057\u307e\u3059\u3002
$$ {\\R} $$<\/div>\u53ef\u7a4d\u5206\u3067\u3042\u308a\u3001\u63a8\u5b9a\u3092\u6e80\u305f\u3059\u3053\u3068<\/p>
$$ {\\forall x\\in \\R,\\quad |f(x)|\\le \\frac{C}{1+x^2}.} $$<\/div><\/dd><\/dl><\/dd><\/dl>

a \u3092<\/i>\u53b3\u5bc6\u306b\u6b63\u306e\u6570<\/a><\/span>\u3068\u3057\u307e\u3059\u3002\u57fa\u672c\u30e2\u30fc\u30c9\u3092\u03c9 0<\/sub> = 2\u03c0 \/ a<\/i><\/span>\u3068\u8868\u3057\u307e\u3059\u3002

$$ {\\hat f} $$<\/div> f<\/i>\u306e\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u6b21\u306e ID \u304c\u5f97\u3089\u308c\u307e\u3059\u3002 <\/p>
$$ { S(t) \\equiv \\sum_{n=-\\infty}^{\\infty} f(t + n a) = \\frac{1}{a} \\sum_{m=-\\infty}^{\\infty} \\hat{f}(m \\omega_0) \\ e^{i m \\omega_0 t} } $$<\/div> \u3002<\/dd><\/dl><\/dd><\/dl>

\u5f0f\u306e\u30c7\u30e2\u30f3\u30b9\u30c8\u30ec\u30fc\u30b7\u30e7\u30f3<\/a><\/span><\/span><\/h3>

\u30dd\u30a2\u30bd\u30f3\u306e\u548c\u306e\u516c\u5f0f\u306e\u5de6\u5074\u306f\u3001\u4e00\u9023\u306e\u9023\u7d9a\u95a2\u6570\u306e\u548c\u3067\u3059\u3002\u7121\u9650\u5927<\/a><\/span>\u306b\u304a\u3051\u308bf<\/i>\u306e\u6319\u52d5\u306b\u95a2\u3057\u3066\u7acb\u3066\u3089\u308c\u305f\u4eee\u8aac\u306f\u3001\u3053\u306e\u7d1a\u6570\u304c\u6b21<\/a><\/span>\u306e\u4efb\u610f\u306e\u30b3\u30f3\u30d1\u30af\u30c8 [-a,a] \u306b\u6b63\u5e38\u306b\u53ce\u675f\u3059\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002

$$ {\\R} $$<\/div> \u3002\u3057\u305f\u304c\u3063\u3066\u3001\u305d\u306e\u548c\u306f\u9023\u7d9a\u95a2\u6570\u3067\u3042\u308a\u3001\u5b9a\u7fa9\u5f0f\u306f\u5468\u671fa<\/i>\u3067\u5468\u671f\u7684\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u307e\u3059\u3002<\/p>

\u3057\u305f\u304c\u3063\u3066\u3001\u305d\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570<\/span>\u306e\u4fc2\u6570\u3092\u8907\u7d20\u6307\u6570\u95a2\u6570\u3067\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002 <\/p>

$$ {c_m=\\int_0^a \\sum_{n\\in\\Z} f(t+na)e^{-2\\mathrm{i}\\pi mt\/a}\\, dt.} $$<\/div><\/dd><\/dl><\/dd><\/dl>

S<\/i>\u3092\u5b9a\u7fa9\u3059\u308b\u7d1a\u6570\u306e\u901a\u5e38\u306e\u53ce\u675f<\/span>\u306b\u3088\u308a\u3001\u7a4d\u5206\u3068\u5408\u8a08\u3092\u4ea4\u63db\u3067\u304d\u308b\u305f\u3081\u3001\u6b21\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002 <\/p>

$$ {c_m=\\sum_{n\\in\\Z}\\int_0^a f(t+na) e^{-2\\mathrm{i}\\pi mt\/a}\\, dt.} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u5404\u7a4d\u5206<\/a><\/span>\u3067\u5909\u6570<\/a><\/span>t<\/i> + na<\/i> = s<\/i>\u306e\u5909\u66f4\u3092\u5b9f\u884c\u3059\u308b\u3068\u3001\u6b21\u306e\u7d50\u679c\u304c\u5f97\u3089\u308c\u307e\u3059\u3002 <\/p>

$$ {c_m=\\sum_{n\\in\\Z}\\int_{na}^{(n+1)a} f(s) e^{-2\\mathrm{i}\\pi m(s-na)\/a}\\, ds=\\hat f(2m\\pi\/a).} $$<\/div><\/dd><\/dl><\/dd><\/dl>

f<\/i>\u3068\u305d\u306e\u5c0e\u95a2\u6570\u306b\u95a2\u3059\u308b\u4eee\u8aac\u3001\u304a\u3088\u3073\u5c0e\u95a2\u6570<\/a><\/span>\u306e\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306b\u95a2\u3059\u308b\u53e4\u5178\u7684\u6052\u7b49\u5f0f\u306b\u3088\u308c\u3070\u3001\u6b21\u306e\u95a2\u6570\u304c\u308f\u304b\u308a\u307e\u3059\u3002

$$ {\\hat f} $$<\/div>\u898b\u7a4d\u3082\u308a\u3092\u78ba\u8a8d\u3059\u308b<\/p>
$$ {\\forall \\omega\\in \\R, \\quad |\\hat f(\\omega)|\\le \\hat C\/(1+\\omega^2).} $$<\/div> \u3002<\/dd><\/dl><\/dd><\/dl>

\u3057\u305f\u304c\u3063\u3066\u3001\u4e00\u9023\u306ecm<\/i>\u306f<\/i><\/sub><\/span>\u7d76\u5bfe\u306b\u53ce\u675f\u3057\u307e\u3059\u3002 S<\/i>\u306e\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u3092\u5408\u8a08\u3067\u304d\u308b\u72b6\u6cc1<\/a><\/span>\u306b\u3042\u308a\u3001\u6b21\u306e\u7d50\u679c\u304c\u5f97\u3089\u308c\u307e\u3059\u3002 <\/p>

$$ {S(t)=\\frac{1}{a}\\sum_{m\\in \\Z}c_m e^{2\\mathrm{i}\\pi mt\/a}=\\frac{1}{a}\\sum_{m\\in\\Z}\\hat f(2m\\pi\/a) e^{2\\mathrm{i}\\pi mt\/a}.} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u3053\u308c\u306f\u3001 2\u03c0 \/ a<\/i><\/span>\u3092\u03c9 0<\/sub><\/span>\u3067\u7f6e\u304d\u63db\u3048\u305f\u6cd5\u3092\u6c42\u3081<\/i>\u305f\u671b\u307e\u3057\u3044\u5f0f\u3067\u3059\u3002<\/p>

\n\"\u30dd\u30a2\u30bd\u30f3\u548c\u306e\u516c\u5f0f<\/figure>

\u4ee3\u66ff\u5354\u5b9a<\/span><\/h3>

\u6b21\u306e\u898f\u5247\u3092\u4f7f\u7528\u3059\u308b\u3068\u3001\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002 <\/p>

$$ { f(x) = \\int_{-\\infty}^{\\infty} \\tilde{F}(\\omega) \\ e^{- \\, 2 \\pi i x \\omega} \\ d \\omega } $$<\/div> \u3001 <\/center>
$$ { \\tilde{F}(\\omega) = \\int_{-\\infty}^{\\infty} f(x) \\ e^{+ \\, 2 \\pi i x \\omega} \\ dx } $$<\/div> \u3001<\/center>

\u6b21\u306b\u3001\u30dd\u30a2\u30bd\u30f3\u306e\u7dcf\u548c\u5f0f\u304c\u66f8\u304d\u63db\u3048\u3089\u308c\u307e\u3059 (

$$ {t=0\\,} $$<\/div>\u305d\u3057\u3066
$$ {a=1\\,} $$<\/div> ): <\/p>
$$ { \\sum_{n \\in \\mathbb{Z}} f(n) \\ = \\ \\sum_{m \\in \\mathbb{Z}} \\tilde{F}(m) } $$<\/div> \u3002<\/center>

\u53ce\u675f\u306e\u6761\u4ef6\u306b\u3064\u3044\u3066<\/span><\/h3>

\u95a2\u6570\u306b\u8ab2\u305b\u3089\u308c\u305f\u898f\u5247\u6027\u6761\u4ef6\u3092\u30aa\u30fc\u30d0\u30fc\u30e9\u30a4\u30c9\u3059\u308b<\/a><\/span>\u5b9f\u7528\u7684\u306a\u65b9\u6cd5

$$ {f\\,} $$<\/div>\u305d\u308c\u306f\u3001\u5206\u5e03\u7406\u8ad6\u306e\u3088\u308a\u4e00\u822c\u7684\u306a\u6587\u8108\u306b\u81ea\u5206\u81ea\u8eab\u3092\u7f6e\u304f\u3053\u3068\u3067\u3059\u3002\u6ce8\u610f\u3059\u308c\u3070
$$ {\\delta (x)\\,} $$<\/div>\u30c7\u30a3\u30e9\u30c3\u30af\u5206\u5e03\u306b\u6b21\u306e\u5206\u5e03\u3092\u5c0e\u5165\u3059\u308b\u3068:<\/p>
<\/center>

\u5408\u8a08\u3092\u518d\u5b9a\u5f0f\u5316\u3059\u308b\u30a8\u30ec\u30ac\u30f3\u30c8\u306a\u65b9\u6cd5\u306f\u3001\u6b21\u306e\u3088\u3046\u306b\u8a00\u3046\u3053\u3068\u3067\u3059\u3002

$$ {\\Delta (x)\\,} $$<\/div>\u306f\u72ec\u81ea\u306e\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3067\u3059\u3002<\/p><\/div>

\u53c2\u8003\u8cc7\u6599<\/h2>
  1. Poissonsche Summenformel \u2013 allemand<\/a><\/li>
  2. Poisson summation formula \u2013 anglais<\/a><\/li>
  3. Formula di sommazione di Poisson \u2013 italien<\/a><\/li>
  4. \u30dd\u30a2\u30bd\u30f3\u548c\u516c\u5f0f \u2013 japonais<\/a><\/li>
  5. F\u00f3rmula do somat\u00f3rio de Poisson \u2013 portugais<\/a><\/li>
  6. \u6cca\u677e\u6c42\u548c\u516c\u5f0f \u2013 chinois<\/a><\/li><\/ol><\/div>\n
    \n

    \n \u30dd\u30a2\u30bd\u30f3\u548c\u306e\u516c\u5f0f – \u5b9a\u7fa9\u30fb\u95a2\u9023\u52d5\u753b\n <\/h2>\n
    \n \n
    \n
    \n