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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 Riesz–Thorin theorem ： ウィキペディア英語版 Riesz–Thorin theorem In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem is a result about ''interpolation of operators''. It is named after Marcel Riesz and his student G. Olof Thorin. This theorem bounds the norms of linear maps acting between spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to which is a Hilbert space, or to and . Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz–Thorin theorem to pass from the simple cases to the complicated cases. The Marcinkiewicz theorem is similar but applies also to a class of non-linear maps. ==Motivation== First we need the following definition: :Definition. Let be two numbers such that . Then for define by: . By splitting up the function in as the product and applying Hölder's inequality to its power, we obtain the following result, foundational in the study of -spaces: :Proposition (log-convexity of -norms). Each satisfies: ::$\backslash |f\backslash |\_\; \backslash leq\; \backslash |f\backslash |\_^\backslash |f\backslash |\_^\backslash theta.$ This result, whose name derives from the convexity of the map on , implies that . On the other hand, if we take the ''layer-cake decomposition'' +  ''f'' 1}}, then we see that ∈ ''L''''p''0}} and ∈ ''L''''p''1}}, whence we obtain the following result: :Proposition. Each in can be written as a sum: , where and . In particular, the above result implies that is included in , the sumset of and in the space of all measurable functions. Therefore, we have the following chain of inclusions: :Corollary. . In practice, we often encounter operators defined on the sumset . For example, the Riemann–Lebesgue lemma shows that the Fourier transform maps boundedly into , and Plancherel's theorem shows that the Fourier transform maps boundedly into itself, whence the Fourier transform $\backslash mathcal$ extends to by setting :$\backslash mathcal(f\_1+f\_2)\; =\; \backslash mathcal\_(f\_1)\; +\; \backslash mathcal\_(f\_2)$ for all and . It is therefore natural to investigate the behavior of such operators on the ''intermediate subspaces'' . To this end, we go back to our example and note that the Fourier transform on the sumset was obtained by taking the sum of two instantiations of the same operator, namely :$\backslash mathcal\_:L^1(\backslash mathbf^d)\; \backslash to\; L^\backslash infty(\backslash mathbf^d),$ :$\backslash mathcal\_:L^2(\backslash mathbf^d)\; \backslash to\; L^2(\backslash mathbf^d).$ These really are the ''same'' operator, in the sense that they agree on the subspace . Since the intersection contains simple functions, it is dense in both and . Densely defined continuous operators admit unique extensions, and so we are justified in considering $\backslash mathcal\_$ and $\backslash mathcal\_$ to be ''the same''. Therefore, the problem of studying operators on the sumset essentially reduces to the study of operators that map two natural domain spaces, and , boundedly to two target spaces: and , respectively. Since such operators map the sumset space to , it is natural to expect that these operators map the intermediate space to the corresponding intermediate space . 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「Riesz–Thorin theorem」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.