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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 Positive harmonic function ： ウィキペディア英語版 Positive harmonic function In mathematics, a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a finite positive measure on the circle. This result, the ''Herglotz representation theorem'', was proved by Gustav Herglotz in 1911. It can be used to give a related formula and characterization for any holomorphic function on the unit disc with positive real part. Such functions had already been characterized in 1907 by Constantin Carathéodory in terms of the positive definiteness of their Taylor coefficients. ==Herglotz representation theorem for harmonic functions== A positive function ''f'' on the unit disk with ''f''(0) = 1 is harmonic if and only if there is a probability measure μ on the unit circle such that :$f(re^)=\backslash int\_0^\; \backslash ,\; d\backslash mu(\backslash varphi).$ The formula clearly defines a positive harmonic function with ''f''(0) = 1. Conversely if ''f'' is positive and harmonic and ''r''''n'' increases to 1, define :$f\_n(z)=f(r\_nz).\; \backslash ,$ Then :$f\_n(re^)\; =\; \backslash int\_0^\; \backslash ,\; f\_n(\backslash varphi)\backslash ,d\backslash phi\; =\backslash int\_0^\; d\backslash mu\_n(\backslash varphi)$ where :$d\backslash mu\_n(\backslash varphi)=\; f(r\_n\; e^)\backslash ,d\backslash varphi$ is a probability measure. By a compactness argument (or equivalently in this case Helly's selection theorem for Stieltjes integrals), a subsequence of these probability measures has a weak limit which is also a probability measure μ. Since ''r''''n'' increases to 1, so that ''f''''n''(''z'') tends to ''f''(''z''), the Herglotz formula follows. 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「Positive harmonic function」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.