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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hardy–Littlewood inequality ： ウィキペディア英語版 Hardy–Littlewood inequality In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if ''f'' and ''g'' are nonnegative measurable real functions vanishing at infinity that are defined on ''n''-dimensional Euclidean space R''n'' then :$\backslash int\_\; f(x)g(x)\; \backslash ,\; dx\; \backslash leq\; \backslash int\_\; f^$ *(x)g^ *(x) \, dx where ''f'' * and ''g'' * are the symmetric decreasing rearrangements of ''f''(''x'') and ''g''(''x''), respectively.〔 〕 ==Proof== From layer cake representation we have:〔〔 :$f(x)=\; \backslash int\_0^\backslash infty\; \backslash chi\_\; \backslash ,\; dr$ :$g(x)=\; \backslash int\_0^\backslash infty\; \backslash chi\_\; \backslash ,\; ds$ where $\backslash chi\_$ denotes the indicator function of the subset ''E'' ''f'' given by :$E\_f=\backslash left\backslash \; \backslash ,$ Analogously, $\backslash chi\_$ denotes the indicator function of the subset ''E'' ''g'' given by :$E\_g=\backslash left\backslash \; \backslash ,$ :$$ \int_ f(x)g(x) \, dx = \displaystyle\int_\int_0^\infty \int_0^\infty \chi_\chi_ \, dr \, ds \, dx :::$=\; \backslash int\_0^\backslash infty\; \backslash int\_0^\backslash infty\; \backslash int\_\backslash chi\_\; \backslash ,\; dx\; \backslash ,\; dr\; \backslash ,\; ds$ :::$=\; \backslash int\_0^\backslash infty\; \backslash int\_0^\backslash infty\; \backslash mu\backslash left(\backslash left\backslash \backslash cap\backslash left\backslash \backslash right)\; \backslash ,\; dr\; \backslash ,\; ds$ :::$\backslash leq\; \backslash int\_0^\backslash infty\; \backslash int\_0^\backslash infty\; \backslash min\backslash left(\backslash mu\backslash left(f(x)>r\backslash right);\backslash mu\backslash left(g(x)>s\backslash right)\backslash right)\; \backslash ,\; dr\; \backslash ,\; ds$ :::$=\; \backslash int\_0^\backslash infty\; \backslash int\_0^\backslash infty\; \backslash min\backslash left(\backslash mu\backslash left(f^$ *(x)>r\right);\mu\left(g^ *(x)>s\right)\right) \, dr \, ds :::$=\; \backslash int\_0^\backslash infty\; \backslash int\_0^\backslash infty\; \backslash mu\backslash left(\backslash left\backslash \backslash cap\backslash left\backslash \backslash right)\; \backslash ,\; dr\; \backslash ,\; ds$ :::$=\; \backslash int\_\; f^$ *(x)g^ *(x) \, dx 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hardy–Littlewood inequality」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.