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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 Birnbaum–Orlicz space ： ウィキペディア英語版 Birnbaum–Orlicz space In the mathematical analysis, and especially in real and harmonic analysis, a Birnbaum–Orlicz space is a type of function space which generalizes the Lp spaces. Like the Lp spaces, they are Banach spaces. The spaces are named for Władysław Orlicz and Zygmunt William Birnbaum, who first defined them in 1931. Besides the ''L''p spaces, a variety of function spaces arising naturally in analysis are Birnbaum–Orlicz spaces. One such space ''L'' log+ ''L'', which arises in the study of Hardy–Littlewood maximal functions, consists of measurable functions ''f'' such that the integral : Here log+ is the positive part of the logarithm. Also included in the class of Birnbaum–Orlicz spaces are many of the most important Sobolev spaces. ==Formal definition== Suppose that μ is a σ-finite measure on a set ''X'', and Φ : [0, ∞) → [0, ∞) is a Young function, i.e., a convex function such that :$\backslash frac\; \backslash to\; \backslash infty,\backslash quad\backslash mathrmx\backslash to\; \backslash infty,$ :$\backslash frac\; \backslash to\; 0,\backslash quad\backslash mathrmx\backslash to\; 0.$ Let $L^\backslash dagger\_\backslash Phi$ be the set of measurable functions ''f'' : ''X'' → R such that the integral :$\backslash int\_X\; \backslash Phi(|f|)\backslash ,\; d\backslash mu$ is finite, where, as usual, functions that agree almost everywhere are identified. This ''may not be'' a vector space (it may fail to be closed under scalar multiplication). The vector space of functions spanned by $L^\backslash dagger\_\backslash Phi$ is the Birnbaum–Orlicz space, denoted $L\_\backslash Phi$. To define a norm on $L\_\backslash Phi$, let Ψ be the Young complement of Φ; that is, :$\backslash Psi(x)\; =\; \backslash int\_0^x\; (\backslash Phi\text{'})^(t)\backslash ,\; dt.$ Note that Young's inequality holds: :$ab\backslash le\; \backslash Phi(a)\; +\; \backslash Psi(b).$ The norm is then given by :$\backslash |f\backslash |\_\backslash Phi\; =\; \backslash sup\backslash left\backslash .$ Furthermore, the space $L\_\backslash Phi$ is precisely the space of measurable functions for which this norm is finite. An equivalent norm is defined on LΦ by :$\backslash |f\backslash |\text{'}\_\backslash Phi\; =\; \backslash inf\backslash left\backslash ,$ and likewise LΦ(μ) is the space of all measurable functions for which this norm is finite. 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「Birnbaum–Orlicz space」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.