{"id":79,"date":"2015-07-08T15:52:09","date_gmt":"2015-07-08T07:52:09","guid":{"rendered":"http:\/\/blog.vanabel.info\/?p=79"},"modified":"2016-12-06T11:48:28","modified_gmt":"2016-12-06T11:48:28","slug":"holderlian-xu-han-shu-de-yi-ge-jian-dan-li-zi","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=79","title":{"rendered":"Holder\u8fde\u7eed\u51fd\u6570\u7684\u4e00\u4e2a\u7b80\u5355\u4f8b\u5b50"},"content":{"rendered":"<p>\u4e3a\u4e86\u65b9\u4fbf\u7406\u89e3Holder\u8fde\u7eed, \u6211\u8fd9\u91cc\u4e3e\u4e00\u4e2a\u4f8b\u5b50.<br \/>\n<div class='latex_examp'><span class='latex_examp_h'>\u4f8b\u5b50 1<\/span><span class='latex_examp_h'>.<\/span> \u4ee4$f(x)=|x|^{\\alpha}$, $1&gt;\\alpha&gt;0$, \u5219$\\|f\\|_{C^\\alpha[-1,1]}=1$.<br \/>\n<\/div><br \/>\n\u4e3a\u4e86\u76f4\u89c2\u8d77\u89c1, \u51fd\u6570$f$\u7684\u56fe\u5f62\u5982\u4e0b<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-85\" src=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2015\/07\/holder_f.png\" alt=\"holder_f\" width=\"360\" height=\"226\" \/><br \/>\n<!--more--><br \/>\n<div class='latex_proof'><span class='latex_proof_h'>\u8bc1\u660e<\/span><span class='latex_proof_h'>.<\/span> \u56de\u5fc6Holder\u6a21\u7684\u5b9a\u4e49<br \/>\n\\[<br \/>\n\\|f\\|_{C^\\alpha[-1,1]}=\\sup_{\\substack{x\\neq y\\\\x,y\\in[-1,1]}}\\frac{|f(x)-f(y)|}{|x-y|^\\alpha}.<br \/>\n\\]<br \/>\n\u4e3a\u4e86\u8ba1\u7b97\u5176holder\u6a21, \u6ce8\u610f\u5230\u51fd\u6570\u4e0e\u5b9a\u4e49\u57df\u7684\u5bf9\u79f0\u6027, \u6211\u4eec\u53ea\u9700\u8003\u8651\u5982\u4e0b\u4e24\u79cd\u60c5\u5f62<br \/>\n<ol><li>$0\\leq x&lt; y$, \u6b64\u65f6\u53ef\u8bbe$x=ky$, $k\\in[0, 1)$ (\u8bf7\u6ce8\u610f\u8fd9\u4e2a\u8bbe\u6cd5\u7684\u826f\u597d\u6027, \u5bf9\u4efb\u610f\u6ee1\u8db3\u6761\u4ef6\u7684$x,y$\u4e00\u5b9a\u5b58\u5728$k$, \u53cd\u8fc7\u6765\u5bf9\u4efb\u610f\u7684$k$\u81f3\u5c11\u5b58\u5728\u4e00\u5bf9\u6ee1\u8db3\u6761\u4ef6\u7684$x,y$). \u56e0\u6b64, \u6211\u4eec\u53ea\u9700\u8003\u8651\u51fd\u6570$g(k,1)$\u5728$[0,1)$\u4e0a\u7684\u6781\u503c, \u5176\u4e2d<br \/>\n\\[<br \/>\ng(x,y)\\eqdef \\frac{|f(x)-f(y)|}{|x-y|^\\alpha}.<br \/>\n\\]<br \/>\n\u5bb9\u6613\u6c42\u5f97$g([0,1))=(0,1]$.<br \/>\n\u4e8b\u5b9e\u4e0a, \u4e3a\u4e86\u76f4\u89c2\u8d77\u89c1, \u6211\u7ed9\u51fa\u6b64\u65f6\u8be5\u51fd\u6570\u7684\u56fe\u8c61\u5982\u4e0b<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-86 size-full\" src=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2015\/07\/holder_g_1.png\" alt=\"holder_g_1\" width=\"360\" height=\"221\" \/><\/li><li>$x&lt; 0\\leq y$\u4e14$|x|\\leq|y|$, \u6b64\u65f6\u53ef\u8bbe$y=kx$, \u4e14$k\\in(-\\infty,-1]$, \u8fd9\u91cc\u7684\u8bbe\u6cd5\u7684\u826f\u597d\u6027\u540c\u524d\u7406\u89e3. \u6c42$g(1,k)$\u5728$(-\\infty,1]$\u7684\u6781\u503c\u5f97\u5230$g((-\\infty,-1])=[0,1)$.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-87 size-full\" src=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2015\/07\/holder_g_2.png\" alt=\"holder_g_2\" width=\"360\" height=\"221\" \/><\/li><\/ol>\u7efc\u4e0a, \u6211\u4eec\u6c42\u5f97\u4e86$|x|^\\alpha$\u5728$[-1,1]$\u4e0a\u7684$C^\\alpha$\u6a21\u662f$1$.<br \/>\n<\/div><br \/>\n<div class='latex_rem'><span class='latex_rem_h'>\u6ce8\u8bb0 1<\/span><span class='latex_rem_h'>.<\/span> \u7c7b\u4f3c\u7684\u8ba8\u8bba, \u5f97\u51fa\u5bf9$|x|^\\alpha$, \u5b83\u5728$[-1,1]$\u4e0a\u7684$C^\\beta$, $\\beta\\neq\\alpha$\u6a21\u4e0d\u5b58\u5728.<br \/>\n<\/div><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4e3a\u4e86\u65b9\u4fbf\u7406\u89e3Holder\u8fde\u7eed, \u6211\u8fd9\u91cc\u4e3e\u4e00\u4e2a\u4f8b\u5b50. \u4f8b\u5b50 1. \u4ee4$f(x)=|x|^{\\alpha}$, $&hellip; <a class=\"more-link\" href=\"https:\/\/blog.vanabel.cn\/?p=79\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">Holder\u8fde\u7eed\u51fd\u6570\u7684\u4e00\u4e2a\u7b80\u5355\u4f8b\u5b50<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[11,12],"class_list":{"0":"post-79","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"hentry","6":"category-math","7":"tag-holder","9":"entry"},"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/79","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=79"}],"version-history":[{"count":2,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/79\/revisions"}],"predecessor-version":[{"id":261,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/79\/revisions\/261"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=79"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=79"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=79"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}