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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 potential ： ウィキペディア英語版 Riesz potential In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space. They generalize to several variables the Riemann–Liouville integrals of one variable. If 0 < α < ''n'', then the Riesz potential ''I''α''f'' of a locally integrable function ''f'' on R''n'' is the function defined by \, \mathrmy|}} where the constant is given by :$c\_\backslash alpha\; =\; \backslash pi^2^\backslash alpha\backslash frac.$ This singular integral is well-defined provided ''f'' decays sufficiently rapidly at infinity, specifically if ''f'' ∈ L''p''(R''n'') with 1 ≤ ''p'' < ''n''/α. If ''p'' > 1, then the rate of decay of ''f'' and that of ''I''α''f'' are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality) :$\backslash |I\_\backslash alpha\; f\backslash |\_\; \backslash le\; C\_p\; \backslash |f\backslash |\_p,\backslash quad\; p^$ *=\frac. More generally, the operators ''I''α are well-defined for complex α such that 0 < Re α < ''n''. The Riesz potential can be defined more generally in a weak sense as the convolution :$I\_\backslash alpha\; f\; =\; f$ *K_\alpha\, where ''K''α is the locally integrable function: :$K\_\backslash alpha(x)\; =\; \backslash frac\backslash frac(\backslash xi)\; =\; |2\backslash pi\backslash xi|^$ and so, by the convolution theorem, :$\backslash widehat(\backslash xi)\; =\; |2\backslash pi\backslash xi|^\; \backslash hat(\backslash xi).$ The Riesz potentials satisfy the following semigroup property on, for instance, rapidly decreasing continuous functions :$I\_\backslash alpha\; I\_\backslash beta\; =\; I\_\backslash$ provided : Furthermore, if 2 < Re α <''n'', then :$\backslash Delta\; I\_\; =\; -I\_\backslash alpha.\backslash$ One also has, for this class of functions, :$\backslash lim\_\; (I^\backslash alpha\; f)(x)\; =\; f(x).$ ==See also== * Bessel potential * Fractional integration * Sobolev space * Fractional Schrödinger equation 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 potential」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.