翻訳と辞書 Words near each other ・ Dirichlet boundary condition ・ Dirichlet character ・ Dirichlet conditions ・ Dirichlet convolution ・ Dirichlet density ・ Dirichlet distribution ・ Dirichlet eigenvalue ・ Dirichlet eta function ・ Dirichlet form ・ Dirichlet integral ・ Dirichlet kernel ・ Dirichlet L-function ・ Dirichlet problem ・ Dirichlet process ・ Dirichlet series ・ Dirichlet space ・ Dirichlet's approximation theorem ・ Dirichlet's energy ・ Dirichlet's principle ・ Dirichlet's test ・ Dirichlet's theorem ・ Dirichlet's theorem on arithmetic progressions ・ Dirichlet's unit theorem ・ Dirichlet-multinomial distribution ・ Dirichletia ・ Dirick Carver ・ Dirico ・ Dirico Airport ・ Diridavumab ・ Diridon Station 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 Dirichlet space ： ウィキペディア英語版 Dirichlet space In mathematics, the Dirichlet space on the domain $\backslash Omega\; \backslash subseteq\; \backslash mathbb,\; \backslash ,\; \backslash mathcal(\backslash Omega)$ (named after Peter Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space $H^2(\backslash Omega)$, for which the ''Dirichlet integral'', defined by :$\backslash mathcal(f)\; :=\; \backslash iint\_\backslash Omega\; |f^\backslash prime(z)|^2\; \backslash ,\; dA\; =\; \backslash iint\_\backslash Omega\; |\backslash partial\_x\; f|^2\; +\; |\backslash partial\_y\; f|^2\; \backslash ,\; dx\; \backslash ,\; dy$ is finite (here ''dA'' denotes the area Lebesgue measure on the complex plane $\backslash mathbb$). The latter is the integral occurring in Dirichlet's principle for harmonic functions. The Dirichlet integral defines a seminorm on $\backslash mathcal(\backslash Omega)$. It is not a norm in general, since $\backslash mathcal(f)\; =\; 0$ whenever ''f'' is a constant function. For $f,\backslash ,\; g\; \backslash in\; \backslash mathcal(\backslash Omega)$, we define :$\backslash mathcal(f,\; \backslash ,\; g)\; :\; =\; \backslash iint\_\backslash Omega\; f\text{'}(z)\; \backslash overline\; \backslash ,\; dA(z).$ This is a semi-inner product, and clearly $\backslash mathcal(f,\; \backslash ,\; f)\; =\; \backslash mathcal(f)$. We may equip $\backslash mathcal(\backslash Omega)$ with an inner product given by :$\backslash langle\; f,\; g\; \backslash rangle\_\; :=\; \backslash langle\; f,\; \backslash ,\; g\; \backslash rangle\_\; +\; \backslash mathcal(f,\; \backslash ,\; g)\; \backslash ;\; \backslash ;\; \backslash ;\; \backslash ;\; \backslash ;\; (f,\; \backslash ,\; g\; \backslash in\; \backslash mathcal(\backslash Omega)),$ where $\backslash langle\; \backslash cdot,\; \backslash ,\; \backslash cdot\; \backslash rangle\_$ is the usual inner product on $H^2\; (\backslash Omega).$ The corresponding norm $\backslash |\; \backslash cdot\; \backslash |\_$ is given by :$\backslash |f\backslash |^2\_\; :=\; \backslash |f\backslash |^2\_\; +\; \backslash mathcal(f)\; \backslash ;\; \backslash ;\; \backslash ;\; \backslash ;\; \backslash ;\; (f\; \backslash in\; \backslash mathcal\; (\backslash Omega)).$ Note that this definition is not unique, another common choice is to take $\backslash |f\backslash |^2\; =\; |f(c)|^2\; +\; \backslash mathcal(f)$, for some fixed $c\; \backslash in\; \backslash Omega$. The Dirichlet space is not an algebra, but the space $\backslash mathcal(\backslash Omega)\; \backslash cap\; H^\backslash infty(\backslash Omega)$ is a Banach algebra, with respect to the norm :$\backslash |f\backslash |\_\; :=\; \backslash |f\backslash |\_\; +\; \backslash mathcal(f)^\; \backslash ;\; \backslash ;\; \backslash ;\; \backslash ;\; \backslash ;\; (f\; \backslash in\; \backslash mathcal(\backslash Omega)\; \backslash cap\; H^\backslash infty(\backslash Omega)).$ We usually have $\backslash Omega\; =\; \backslash mathbb$ (the unit disk of the complex plane $\backslash mathbb$), in that case $\backslash mathcal(\backslash mathbb):=\backslash mathcal$, and if :$f(z)\; =\; \backslash sum\_\; a\_n\; z^n\; \backslash ;\; \backslash ;\; \backslash ;\; \backslash ;\; \backslash ;\; (f\; \backslash in\; \backslash mathcal),$ then :$D(f)\; =\backslash sum\_\; n\; |a\_n|^2,$ and :$\backslash |f\; \backslash |^2\_\backslash mathcal\; =\; \backslash sum\_\; (n+1)\; |a\_n|^2.$ Clearly, $\backslash mathcal$ contains all the polynomials and, more generally, all functions $f$, holomorphic on $\backslash mathbb$ such that $f\text{'}$ is bounded on $\backslash mathbb$. The reproducing kernel of $\backslash mathcal$ at $w\; \backslash in\; \backslash mathbb\; \backslash setminus\; \backslash$ is given by :$k\_w(z)\; =\; \backslash frac\; \backslash setminus\; \backslash ).$ ==See also== *Banach space *Bergman space *Hardy space *Hilbert space 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「Dirichlet 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.