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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 Besov measure ： ウィキペディア英語版 Besov measure In mathematics — specifically, in the fields of probability theory and inverse problems — Besov measures and associated Besov-distributed random variables are generalisations of the notions of Gaussian measures and random variables, Laplace distributions, and other classical distributions. They are particularly useful in the study of inverse problems on function spaces for which a Gaussian Bayesian prior is an inappropriate model. The construction of a Besov measure is similar to the construction of a Besov space, hence the nomenclature. == Definitions == Let $H$ be a separable Hilbert space of functions defined on a domain $D\; \backslash subseteq\; \backslash mathbb^$, and let $\backslash \; \backslash \}$ be a complete orthonormal basis for $H$. Let $s\; \backslash in\; \backslash mathbb$ and . For $u\; =\; \backslash sum\_\; e\_\; \backslash in\; H$, define :$\backslash |\; u\; \backslash |\_\}\; u\_\; e\_\; \backslash right\backslash |\_^\; n^\; +\; \backslash frac\; -\; 1)\}\; |\; u\_\; |^\; \backslash right)^.$ This defines a norm on the subspace of $H$ for which it is finite, and we let $X^$ denote the completion of this subspace with respect to this new norm. The motivation for these definitions arises from the fact that $\backslash |\; u\; \backslash |\_^(D)$. Let $\backslash kappa\; >\; 0$ be a scale parameter, similar to the precision (the reciprocal of the variance) of a Gaussian measure. We now define a $X^$-valued random variable $u$ by :$u\; :=\; \backslash sum\_\; +\; \backslash frac\; -\; \backslash frac)\}\; \backslash kappa^\}\; \backslash xi\_\; e\_,$ where $\backslash xi\_,\; \backslash xi\_,\; \backslash dots$ are sampled independently and identically from the generalized Gaussian measure on $\backslash mathbb$ with Lebesgue probability density function proportional to $\backslash exp\; (\; -\; \backslash tfrac\; |\; \backslash xi\_\; |^\; )$. Informally, $u$ can be said to have a probability density function proportional to $\backslash exp\; (-\; \backslash tfrac\; \backslash |\; u\; \backslash |\_\; )$ with respect to infinite-dimensional Lebesgue measure (which does not make rigorous sense), and is therefore a natural candidate for a “typical” element of $X^$. Nevertheless, it can be shown that the series defining $u$ converges in $X^$ almost surely, and therefore gives a well-defined $X^$-valued random variable. The random variable $u$ is said to be Besov distributed with parameters $\backslash kappa,\; s,\; p$, and the induced probability measure on $X^$ is called a Besov measure. 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「Besov measure」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.