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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 transform ： ウィキペディア英語版 Riesz transform In the mathematical theory of harmonic analysis, the Riesz transforms are a family of generalizations of the Hilbert transform to Euclidean spaces of dimension ''d'' > 1. They are a type of singular integral operator, meaning that they are given by a convolution of one function with another function having a singularity at the origin. Specifically, the Riesz transforms of a complex-valued function ƒ on R''d'' are defined by \,dt|}} for ''j'' = 1,2,...,''d''. The constant ''c''''d'' is a dimensional normalization given by :$c\_d\; =\; \backslash frac\{|x|^\{d+1\}\}.$ The Riesz transforms arises in the study of differentiability properties of harmonic potentials in potential theory and harmonic analysis. In particular, they arise in the proof of the Calderón-Zygmund inequality . ==Multiplier properties== The Riesz transforms are given by a Fourier multiplier. Indeed, the ) = \sum_^d \rho_ R_kf. These three properties in fact characterize the Riesz transform in the following sense. Let ''T''=(''T''''1'',…,''T''''d'') be a ''d''-tuple of bounded linear operators from ''L''2(R''d'') to ''L''2(R''d'') such that * ''T'' commutes with all dilations and translations. * ''T'' is equivariant with respect to rotations. Then, for some constant ''c'', ''T'' = ''cR''. 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 transform」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.