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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 Szegő kernel ： ウィキペディア英語版 Szegő kernel In the mathematical study of several complex variables, the Szegő kernel is an integral kernel that gives rise to a reproducing kernel on a natural Hilbert space of holomorphic functions. It is named for its discoverer, the Hungarian mathematician Gábor Szegő. Let Ω be a bounded domain in C''n'' with ''C''2 boundary, and let ''A''(Ω) denote the set of all holomorphic functions in Ω that are continuous on $\backslash overline$. Define the Hardy space ''H''2(∂Ω) to be the closure in ''L''2(∂Ω) of the restrictions of elements of ''A''(Ω) to the boundary. The Poisson integral implies that each element ''ƒ'' of ''H''2(∂Ω) extends to a holomorphic function ''Pƒ'' in Ω. Furthermore, for each ''z'' ∈ Ω, the map :$f\backslash mapsto\; Pf(z)$ defines a continuous linear functional on ''H''2(∂Ω). By the Riesz representation theorem, this linear functional is represented by a kernel ''k''''z'', which is to say :$Pf(z)\; =\; \backslash int\_\; f(\backslash zeta)\backslash overline\backslash ,d\backslash sigma(\backslash zeta).$ The Szegő kernel is defined by :$S(z,\backslash zeta)\; =\; \backslash overline,\backslash quad\; z\backslash in\backslash Omega,\backslash zeta\backslash in\backslash partial\backslash Omega.$ Like its close cousin, the Bergman kernel, the Szegő kernel is holomorphic in ''z''. In fact, if ''φ''''i'' is an orthonormal basis of ''H''2(∂Ω) consisting entirely of the restrictions of functions in ''A''(Ω), then a Riesz–Fischer theorem argument shows that :$S(z,\backslash zeta)\; =\; \backslash sum\_^\backslash infty\; \backslash phi\_i(z)\backslash overline.$ ==References== * 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「Szegő 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.