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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 Babenko–Beckner inequality ： ウィキペディア英語版 Babenko–Beckner inequality In mathematics, the Babenko–Beckner inequality (after K. Ivan Babenko and William E. Beckner) is a sharpened form of the Hausdorff–Young inequality having applications to uncertainty principles in the Fourier analysis of Lp spaces. The (''q'', ''p'')-norm of the ''n''-dimensional Fourier transform is defined to be〔Iwo Bialynicki-Birula. ''Formulation of the uncertainty relations in terms of the Renyi entropies.'' (arXiv:quant-ph/0608116v2 )〕 : In 1961, Babenko〔K.I. Babenko. ''An ineqality in the theory of Fourier analysis.'' Izv. Akad. Nauk SSSR, Ser. Mat. 25 (1961) pp. 531–542 English transl., Amer. Math. Soc. Transl. (2) 44, pp. 115–128〕 found this norm for ''even'' integer values of ''q''. Finally, in 1975, using Hermite functions as eigenfunctions of the Fourier transform, Beckner〔W. Beckner, ''Inequalities in Fourier analysis.'' Annals of Mathematics, Vol. 102, No. 6 (1975) pp. 159–182.〕 proved that the value of this norm for all $q\; \backslash ge\; 2$ is :$\backslash |\backslash mathcal\; F\backslash |\_\; =\; \backslash left(p^/q^\backslash right)^.$ Thus we have the Babenko–Beckner inequality that :$\backslash |\backslash mathcal\; Ff\backslash |\_q\; \backslash le\; \backslash left(p^/q^\backslash right)^\; \backslash |f\backslash |\_p.$ To write this out explicitly, (in the case of one dimension,) if the Fourier transform is normalized so that :$g(y)\; \backslash approx\; \backslash int\_\; e^\; f(x)\backslash ,dx\backslash textf(x)\; \backslash approx\; \backslash int\_\; e^\; g(y)\backslash ,dy,$ then we have :$\backslash left(\backslash int\_\; |g(y)|^q\; \backslash ,dy\backslash right)^\; \backslash le\; \backslash left(p^/q^\backslash right)^\; \backslash left(\backslash int\_\; |f(x)|^p\; \backslash ,dx\backslash right)^$ or more simply :$\backslash left(\backslash sqrt\; q\; \backslash int\_\; |g(y)|^q\; \backslash ,dy\backslash right)^$ \le \left(\sqrt p \int_ |f(x)|^p \,dx\right)^. ==Main ideas of proof== Throughout this sketch of a proof, let : (Except for ''q'', we will more or less follow the notation of Beckner.) 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「Babenko–Beckner 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.