翻訳と辞書 Words near each other ・ Cameroon–China relations ・ Cameroon–Nigeria relations ・ Cameroon–Russia relations ・ Cameroon–United States relations ・ Camerota ・ Cameroun Express ・ Camerton ・ Camerton (band) ・ Camerton (LNWR) railway station ・ Camerton Court ・ Camerton, Cumbria ・ Camerton, East Riding of Yorkshire ・ Camerton, Somerset ・ Camerunadora ・ Cameron–Erdős conjecture ・ Cameron–Martin theorem ・ Cameroon ・ Cameroon Academy Awards ・ Cameroon Air Force ・ Cameroon Airlines ・ Cameroon Airlines Flight 3701 ・ Cameroon Airlines Flight 786 ・ Cameroon at the 1964 Summer Olympics ・ Cameroon at the 1968 Summer Olympics ・ Cameroon at the 1972 Summer Olympics ・ Cameroon at the 1976 Summer Olympics ・ Cameroon at the 1980 Summer Olympics ・ Cameroon at the 1984 Summer Olympics ・ Cameroon at the 1988 Summer Olympics ・ Cameroon at the 1992 Summer Olympics Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cameron–Martin theorem ： ウィキペディア英語版 Cameron–Martin theorem In mathematics, the Cameron–Martin theorem or Cameron–Martin formula (named after Robert Horton Cameron and W. T. Martin) is a theorem of measure theory that describes how abstract Wiener measure changes under translation by certain elements of the Cameron–Martin Hilbert space. ==Motivation== The standard Gaussian measure γ''n'' on ''n''-dimensional Euclidean space R''n'' is not translation-invariant. (In fact, there is a unique translation invariant Radon measure up to scale by Haar's theorem: the ''n''-dimensional Lebesgue measure, denoted here ''dx''.) Instead, a measurable subset ''A'' has Gaussian measure :$\backslash gamma\_n(A)\; =\; \backslash frac\backslash right)\backslash ,dx.$ Here $\backslash langle\; x,x\backslash rangle\_$ refers to the standard Euclidean dot product in R''n''. The Gaussian measure of the translation of ''A'' by a vector ''h'' ∈ R''n'' is :$\backslash begin$ \gamma_n(A-h) &= \frac\right)\,dx\\ &=\frac - \langle h, h\rangle_}\right)\exp\left(-\tfrac12\langle x, x\rangle_\right)\,dx. \end So under translation through ''h'', the Gaussian measure scales by the distribution function appearing in the last display: :$\backslash exp\backslash left(\backslash frac\}\backslash right)=\backslash exp\backslash left(\backslash langle\; x,\; h\backslash rangle\_\; -\; \backslash tfrac12\backslash |h\backslash |\_^\backslash right).$ The measure that associates to the set ''A'' the number γ''n''(''A''−''h'') is the pushforward measure, denoted (''T''''h'')∗(γ''n''). Here ''T''''h'' : R''n'' → R''n'' refers to the translation map: ''T''''h''(''x'') = ''x'' + ''h''.. The above calculation shows that the Radon–Nikodym derivative of the pushforward measure with respect to the original Gaussian measure is given by :$\backslash frac)\_\; (\backslash gamma^)\}\}\; (x)\; =\; \backslash exp\; \backslash left(\; \backslash left\; \backslash langle\; h,\; x\; \backslash right\; \backslash rangle\_\}\; -\; \backslash tfrac\; \backslash |\; h\; \backslash |\_\}^\; \backslash right).$ Abstract Wiener measure ''γ'' on a separable Banach space ''E'', where ''i'' : ''H'' → ''E'' is an abstract Wiener space, is also a "Gaussian measure" in a suitable sense. How does it change under translation? It turns out that a similar formula to the one above holds if we consider only translations by elements of the dense subspace ''i''(''H'') ⊆ ''E''. 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「Cameron–Martin theorem」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.