{"id":39603,"date":"2023-09-27T14:43:37","date_gmt":"2023-09-27T06:43:37","guid":{"rendered":"https:\/\/www.techphant.cn\/?p=39603"},"modified":"2023-09-27T14:43:37","modified_gmt":"2023-09-27T06:43:37","slug":"bdljxhsdjhxz","status":"publish","type":"post","link":"https:\/\/www.techphant.cn\/blog\/39603.html","title":{"rendered":"Bochner\u5b9a\u7406\uff1a\u89e3\u6790\u51fd\u6570\u7684\u51e0\u4f55\u6027\u8d28"},"content":{"rendered":"

  Bochner\u5b9a\u7406\u662f\u4e00\u4e2a\u5173\u4e8e\u89e3\u6790\u51fd\u6570\u7684\u51e0\u4f55\u6027\u8d28\u7684\u5b9a\u7406\uff0c\u5b83\u662f20\u4e16\u7eaa\u6570\u5b66\u5bb6Salomon Bochner\u57281932\u5e74\u53d1\u8868\u7684\u4e00\u7bc7\u8bba\u6587\u4e2d\u63d0\u51fa\u7684\u3002\u5b83\u6307\u51fa\uff0c\u4e00\u4e2a\u89e3\u6790\u51fd\u6570\u7684\u5fae\u5206\u5f62\u5f0f\u53ef\u4ee5\u7528\u5b83\u7684Fourier\u7cfb\u6570\u6765\u8868\u793a\uff0c\u800cFourier\u7cfb\u6570\u53c8\u53ef\u4ee5\u7528\u5b83\u7684\u51e0\u4f55\u6027\u8d28\u6765\u8868\u793a\u3002Bochner\u5b9a\u7406\u6709\u52a9\u4e8e\u6211\u4eec\u7406\u89e3\u89e3\u6790\u51fd\u6570\u7684\u51e0\u4f55\u6027\u8d28\uff0c\u4ece\u800c\u66f4\u597d\u5730\u5206\u6790\u89e3\u6790\u51fd\u6570\u7684\u7279\u6027\u3002\u672c\u6587\u5c06\u4ecb\u7ecdBochner\u5b9a\u7406\uff0c\u5e76\u7ed3\u5408\u5b9e\u4f8b\u6df1\u5165\u8ba8\u8bba\u5b83\u7684\u5e94\u7528\u3002<\/p>\n

Bochner\u5b9a\u7406\u7684\u5b9a\u4e49<\/h2>\n

  Bochner\u5b9a\u7406\u6307\u51fa\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u89e3\u6790\u51fd\u6570f(x)\uff0c\u5b83\u7684Fourier\u7cfb\u6570\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n

  c(n)=1\/2\u03c0\u222bf(x)e-inx<\/sup>dx<\/p>\n

  \u5176\u4e2d\uff0cn\u662f\u4e00\u4e2a\u6574\u6570\uff0c\u8868\u793a\u51fd\u6570f(x)\u7684\u9891\u7387\u3002Bochner\u5b9a\u7406\u8fd8\u6307\u51fa\uff0c\u8fd9\u4e9bFourier\u7cfb\u6570c(n)\u53ef\u4ee5\u7528\u5b83\u4eec\u7684\u51e0\u4f55\u6027\u8d28\u6765\u8868\u793a\uff1a<\/p>\n

  c(n)=1\/2\u03c0\u222bf(x)e-inx<\/sup>dx=\u2211k=-\u221e<\/sub>\u221e<\/sup>f(xk<\/sub>)e-inxk<\/sub><\/sup>dx<\/p>\n

  \u8fd9\u91cc\uff0cxk<\/sub>\u8868\u793a\u51fd\u6570f(x)\u7684\u96f6\u70b9\uff0c\u8fd9\u4e9b\u96f6\u70b9\u662f\u51fd\u6570f(x)\u7684\u51e0\u4f55\u6027\u8d28\u7684\u5173\u952e\u3002\u56e0\u6b64\uff0cBochner\u5b9a\u7406\u53ef\u4ee5\u7528\u6765\u8868\u793a\u4e00\u4e2a\u89e3\u6790\u51fd\u6570\u7684\u51e0\u4f55\u6027\u8d28\uff0c\u901a\u8fc7\u5b83\uff0c\u6211\u4eec\u53ef\u4ee5\u66f4\u597d\u5730\u5206\u6790\u89e3\u6790\u51fd\u6570\u7684\u7279\u6027\u3002<\/p>\n

Bochner\u5b9a\u7406\u7684\u5e94\u7528<\/h2>\n

  \u5e94\u75281\uff1a\u6c42\u89e3\u5e38\u5fae\u5206\u65b9\u7a0b<\/h3>\n

  Bochner\u5b9a\u7406\u53ef\u4ee5\u7528\u6765\u6c42\u89e3\u5e38\u5fae\u5206\u65b9\u7a0b\u3002\u4f8b\u5982\uff0c\u8003\u8651\u4e00\u4e2a\u7b80\u5355\u7684\u4e8c\u9636\u5e38\u5fae\u5206\u65b9\u7a0b\uff1a<\/p>\n

  y”+ay'+by=0<\/p>\n

  \u5c06\u4e0a\u5f0f\u6539\u5199\u4e3a\u89e3\u6790\u51fd\u6570\u7684\u5f62\u5f0f\uff1a<\/p>\n

  y=Acos(\u03c9x)+Bsin(\u03c9x)<\/p>\n

  \u7531Bochner\u5b9a\u7406\uff0c\u53ef\u4ee5\u77e5\u9053\uff1a<\/p>\n

  \u03c9=\u221a(a2<\/sup>+4b)<\/p>\n

  \u56e0\u6b64\uff0c\u53ef\u4ee5\u6839\u636eBochner\u5b9a\u7406\u6c42\u89e3\u4e0a\u8ff0\u5e38\u5fae\u5206\u65b9\u7a0b\u3002<\/p>\n

  \u5e94\u75282\uff1a\u8ba1\u7b97\u590d\u6570\u51fd\u6570\u7684\u8c31\u5bc6\u5ea6<\/h3>\n

  Bochner\u5b9a\u7406\u8fd8\u53ef\u4ee5\u7528\u6765\u8ba1\u7b97\u590d\u6570\u51fd\u6570\u7684\u8c31\u5bc6\u5ea6\u3002\u4f8b\u5982\uff0c\u8003\u8651\u4e00\u4e2a\u590d\u6570\u51fd\u6570\uff1a<\/p>\n

  f(x)=Acos(\u03c9x)+Bsin(\u03c9x)<\/p>\n

  \u7531Bochner\u5b9a\u7406\uff0c\u53ef\u4ee5\u77e5\u9053\uff1a<\/p>\n

  f(x)\u7684\u8c31\u5bc6\u5ea6\u4e3a\uff1a<\/p>\n

  P(\u03c9)=|A|2<\/sup>+|B|2<\/sup><\/p>\n

  \u56e0\u6b64\uff0c\u53ef\u4ee5\u6839\u636eBochner\u5b9a\u7406\u8ba1\u7b97\u590d\u6570\u51fd\u6570\u7684\u8c31\u5bc6\u5ea6\u3002<\/p>\n

\u603b\u7ed3<\/h2>\n

  Bochner\u5b9a\u7406\u662f\u4e00\u4e2a\u5173\u4e8e\u89e3\u6790\u51fd\u6570\u7684\u51e0\u4f55\u6027\u8d28\u7684\u5b9a\u7406\uff0c\u5b83\u6307\u51fa\uff0c\u4e00\u4e2a\u89e3\u6790\u51fd\u6570\u7684\u5fae\u5206\u5f62\u5f0f\u53ef\u4ee5\u7528\u5b83\u7684Fourier\u7cfb\u6570\u6765\u8868\u793a\uff0c\u800cFourier\u7cfb\u6570\u53c8\u53ef\u4ee5\u7528\u5b83\u7684\u51e0\u4f55\u6027\u8d28\u6765\u8868\u793a\u3002<\/strong>Bochner\u5b9a\u7406\u6709\u52a9\u4e8e\u6211\u4eec\u7406\u89e3\u89e3\u6790\u51fd\u6570\u7684\u51e0\u4f55\u6027\u8d28\uff0c\u4ece\u800c\u66f4\u597d\u5730\u5206\u6790\u89e3\u6790\u51fd\u6570\u7684\u7279\u6027\u3002\u5b83\u53ef\u4ee5\u7528\u6765\u6c42\u89e3\u5e38\u5fae\u5206\u65b9\u7a0b\uff0c\u4e5f\u53ef\u4ee5\u7528\u6765\u8ba1\u7b97\u590d\u6570\u51fd\u6570\u7684\u8c31\u5bc6\u5ea6\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

Bochner\u5b9a\u7406\u662f\u4e00\u4e2a\u5173\u4e8e\u89e3\u6790\u51fd\u6570\u7684\u51e0\u4f55\u6027\u8d28\u7684\u5b9a\u7406\uff0c\u5b83\u662f20\u4e16\u7eaa\u6570\u5b66\u5bb6SalomonBochner\u57281932\u5e74\u53d1\u8868\u7684\u4e00\u7bc7\u8bba\u6587\u4e2d\u63d0\u51fa\u7684\u3002\u5b83\u6307\u51fa\uff0c\u4e00\u4e2a\u89e3\u6790\u51fd\u6570\u7684\u5fae\u5206\u5f62\u5f0f\u53ef<\/p>\n","protected":false},"author":1,"featured_media":4132,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","footnotes":""},"categories":[36],"tags":[305],"class_list":["post-39603","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-blog","tag-dzjs"],"_links":{"self":[{"href":"https:\/\/www.techphant.cn\/wp-json\/wp\/v2\/posts\/39603","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.techphant.cn\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.techphant.cn\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.techphant.cn\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.techphant.cn\/wp-json\/wp\/v2\/comments?post=39603"}],"version-history":[{"count":0,"href":"https:\/\/www.techphant.cn\/wp-json\/wp\/v2\/posts\/39603\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.techphant.cn\/wp-json\/wp\/v2\/media\/4132"}],"wp:attachment":[{"href":"https:\/\/www.techphant.cn\/wp-json\/wp\/v2\/media?parent=39603"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.techphant.cn\/wp-json\/wp\/v2\/categories?post=39603"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.techphant.cn\/wp-json\/wp\/v2\/tags?post=39603"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}