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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 Bergman space ： ウィキペディア英語版 Bergman space In complex analysis, functional analysis and operator theory, a Bergman space is a function space of holomorphic functions in a domain ''D'' of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, for , the Bergman space is the space of all holomorphic functions $f$ in ''D'' for which the p-norm is finite: : The quantity $\backslash |f\backslash |\_$ is called the ''norm'' of the function ; it is a true norm if $p\; \backslash geq\; 1$. Thus is the subspace of holomorphic functions that are in the space L''p''(''D''). The Bergman spaces are Banach spaces, which is a consequence of the estimate, valid on compact subsets ''K'' of ''D'': Thus convergence of a sequence of holomorphic functions in implies also compact convergence, and so the limit function is also holomorphic. If , then is a reproducing kernel Hilbert space, whose kernel is given by the Bergman kernel. ==Special cases and generalisations== If the domain is bounded, then the norm is often given by :$\backslash |f\backslash |\_\; :=\; \backslash left(\backslash int\_D\; |f(z)|^p\backslash ,dA\backslash right)^\; \backslash ;\; \backslash ;\; \backslash ;\; \backslash ;\; \backslash ;\; (f\; \backslash in\; A^p(D)),$ where $A$ is a normalised Lebesgue measure of the complex plane, i.e. The Bergman space is usually defined on the open unit disk $\backslash mathbb$ of the complex plane, in which case $A^p(\backslash mathbb):=A^p$. In the Hilbert space case, given $f(z)=\; \backslash sum\_^\backslash infty\; a\_n\; z^n\; \backslash in\; A^2$, we have :$\backslash |f\backslash |^2\_\; :=\; \backslash frac\; \backslash int\_\backslash mathbb\; |f(z)|^2\; \backslash ,\; dz\; =\; \backslash sum\_^\backslash infty\; \backslash frac,$ that is, is isometrically isomorphic to the weighted ''ℓp(1/(n+1))'' space.〔 In particular the polynomials are dense in . Similarly, if , the right (or the upper) complex half-plane, then :$\backslash |F\backslash |^2\_\; :=\; \backslash frac\; \backslash int\_\; |F(z)|^2\; \backslash ,\; dz\; =\; \backslash int\_0^\backslash infty\; |f(t)|^2\backslash frac,$ where $F(z)=\; \backslash int\_0^\backslash infty\; f(t)e^\; \backslash ,\; dt$, that is, is isometrically isomorphic to the weighted ''Lp1/t (0,∞)'' space (via the Laplace transform).〔〔 The weighted Bergman space is defined in an analogous way,〔 i.e. :$\backslash |f\backslash |\_\; :=\; \backslash left(\; \backslash int\_D\; |f(x+iy)|^2\; \backslash ,\; w(x+iy)\; \backslash ,\; dx\; \backslash ,\; dy\; \backslash right)^,$ provided that is chosen in such way, that $A^p\_w(D)$ is a Banach space (or a Hilbert space, if ). In case where $D=\; \backslash mathbb$, by a weighted Bergman space $A^p\_\backslash alpha$〔 we mean the space of all analytic functions such that : and similarly on the right half-plane (i.e. $A^p\_\backslash alpha(\backslash mathbb\_+)$) we have〔 :$\backslash |f\backslash |\_\; :=\; \backslash left(\; \backslash frac\backslash int\_\; |f(x+iy)|^p\; x^\backslash alpha\; \backslash ,\; dx\; \backslash ,\; dy\; \backslash right)^,$ and this space is isometrically isomorphic, via the Laplace transform, to the space $L^2(\backslash mathbb\_+,\; \backslash ,\; d\backslash mu\_\backslash alpha)$,〔〔 where :$d\backslash mu\_\backslash alpha\; :=\; \backslash frac\_+\}$, that is : 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「Bergman 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.