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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 Hausdorff–Young inequality ： ウィキペディア英語版 Hausdorff–Young inequality In mathematics, the Hausdorff−Young inequality bounds the ''L''''q''-norm of the Fourier coefficients of a periodic function for ''q'' ≥ 2. proved the inequality for some special values of ''q'', and proved it in general. More generally the inequality also applies to the Fourier transform of a function on a locally compact group, such as R''n'', and in this case and gave a sharper form of it called the Babenko–Beckner inequality. We consider the Fourier operator, namely let ''T'' be the operator that takes a function $f$ on the unit circle and outputs the sequence of its Fourier coefficients : $\backslash widehat(n)=\backslash frac\backslash int\_0^e^f(x)\backslash ,dx,\backslash quad\; n=0,\backslash pm1,\backslash pm2,\backslash dots.$ Parseval's theorem shows that ''T'' is bounded from $L^2$ to $\backslash ell^2$ with norm 1. On the other hand, clearly, :$|(Tf)(n)|=|\backslash widehat(n)|=\backslash left|\backslash frac\backslash int\_0^e^f(t)\backslash ,dt\backslash right|\backslash leq\; \backslash frac\; \backslash int\_0^|f(t)|\backslash ,dt$ so ''T'' is bounded from $L^1$ to $\backslash ell^\backslash infty$ with norm 1. Therefore we may invoke the Riesz–Thorin theorem to get, for any 1 < ''p'' < 2 that ''T'', as an operator from $L^p$ to $\backslash ell^q$, is bounded with norm 1, where :$\backslash frac+\backslash frac=1.$ In a short formula, this says that :$\backslash left(\backslash sum\_^\backslash infty\; |\backslash widehat(n)|^q\backslash right)^\backslash leq$ \left( \frac\int_0^|f(t)|^p\,dt\right)^. This is the well known Hausdorff–Young inequality. For ''p'' > 2 the natural extrapolation of this inequality fails, and the fact that a function belongs to $L^p$, does not give any additional information on the order of growth of its Fourier series beyond the fact that it is in $\backslash ell^2$. ==Optimal estimates== The constant involved in the Hausdorff–Young inequality can be made optimal by using careful estimates from the theory of harmonic analysis. If $f\backslash in\; L^p$ for :$\backslash |\backslash hat\backslash |\_\backslash leq\; p^q^\backslash |f\backslash |\_$ where $q=p/(p-1)$ is the Hölder conjugate of $p$ 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「Hausdorff–Young 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.