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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 Hölder condition ： ウィキペディア英語版 Hölder condition In mathematics, a real or complex-valued function ''f'' on ''d''-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants ''C'', α, such that : $|\; f(x)\; -\; f(y)\; |\; \backslash leq\; C\backslash parallel\; x\; -\; y\backslash parallel^$ for all ''x'' and ''y'' in the domain of ''f''. More generally, the condition can be formulated for functions between any two metric spaces. The number α is called the ''exponent'' of the Hölder condition. If α = 1, then the function satisfies a Lipschitz condition. If α = 0, then the function simply is bounded. The condition is named after Otto Hölder. We have the following chain of inclusions for functions over a compact subset of the real line : Continuously differentiable ⊆Lipschitz continuous ⊆ α-Hölder continuous ⊆ uniformly continuous ⊆ continuous where 0 < α ≤1. ==Hölder spaces== Hölder spaces consisting of functions satisfying a Hölder condition are basic in areas of functional analysis relevant to solving partial differential equations, and in dynamical systems. The Hölder space ''C''''k'',α(Ω), where Ω is an open subset of some Euclidean space and ''k'' ≥ 0 an integer, consists of those functions on Ω having continuous derivatives up to order ''k'' and such that the ''k''th partial derivatives are Hölder continuous with exponent α, where 0 < α ≤ 1. This is a locally convex topological vector space. If the Hölder coefficient :$|\; f\; |\_\; \backslash frac,$ is finite, then the function ''f'' is said to be ''(uniformly) Hölder continuous with exponent α in Ω.'' In this case, the Hölder coefficient serves as a seminorm. If the Hölder coefficient is merely bounded on compact subsets of Ω, then the function ''f'' is said to be ''locally Hölder continuous with exponent α in Ω.'' If the function ''f'' and its derivatives up to order ''k'' are bounded on the closure of Ω, then the Hölder space $C^(\backslash overline)$ can be assigned the norm :$\backslash |\; f\; \backslash |\_+\backslash max\_\; \backslash left\; |\; D^\backslash beta\; f\; \backslash right\; |\_\; =\; \backslash max\_\; \backslash sup\_\; \backslash left\; |D^\backslash beta\; f\; (x)\; \backslash right\; |.$ These norms and seminorms are often denoted simply $|\; f\; |\_$ and $\backslash |\; f\; \backslash |\_$ or also $|\; f\; |\_\backslash ;$ and $\backslash |\; f\; \backslash |\_$ in order to stress the dependence on the domain of ''f''. If Ω is open and bounded, then $C^(\backslash overline)$ is a Banach space with respect to the norm $\backslash |\backslash cdot\backslash |\_\{C^\{k,\; \backslash alpha\}\}$. 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「Hölder condition」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.