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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 kernel ： ウィキペディア英語版 Bergman kernel In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is a reproducing kernel for the Hilbert space of all square integrable holomorphic functions on a domain ''D'' in C''n''. In detail, let L2(''D'') be the Hilbert space of square integrable functions on ''D'', and let ''L''2,''h''(''D'') denote the subspace consisting of holomorphic functions in ''D'': that is, :$L^(D)\; =\; L^2(D)\backslash cap\; H(D)$ where ''H''(''D'') is the space of holomorphic functions in ''D''. Then ''L''2,''h''(''D'') is a Hilbert space: it is a closed linear subspace of ''L''2(''D''), and therefore complete in its own right. This follows from the fundamental estimate, that for a holomorphic square-integrable function ''ƒ'' in ''D'' for every compact subset ''K'' of ''D''. Thus convergence of a sequence of holomorphic functions in ''L''2(''D'') implies also compact convergence, and so the limit function is also holomorphic. Another consequence of () is that, for each ''z'' ∈ ''D'', the evaluation :$\backslash operatorname\_z\; :\; f\backslash mapsto\; f(z)$ is a continuous linear functional on ''L''2,''h''(''D''). By the Riesz representation theorem, this functional can be represented as the inner product with an element of ''L''2,''h''(''D''), which is to say that :$\backslash operatorname\_z\; f\; =\; \backslash int\_D\; f(\backslash zeta)\backslash overline\backslash ,d\backslash mu(\backslash zeta).$ The Bergman kernel ''K'' is defined by :$K(z,\backslash zeta)\; =\; \backslash overline.$ The kernel ''K''(''z'',ζ) holomorphic in ''z'' and antiholomorphic in ζ, and satisfies :$f(z)\; =\; \backslash int\_D\; K(z,\backslash zeta)f(\backslash zeta)\backslash ,d\backslash mu(\backslash zeta).$ One key observation about this picture is that ''L''2,''h''(''D'') may be identified with the space of $L^2$ holomorphic (n,0)-forms on D, via multiplication by $dz^1\backslash wedge\; \backslash cdots\; \backslash wedge\; dz^n$. Since the $L^2$ inner product on this space is manifestly invariant under biholomorphisms of D, the Bergman kernel and the associated Bergman metric are therefore automatically invariant under the automorphism group of the domain. ==See also== * Bergman metric * Bergman space * Szegő kernel 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 kernel」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.