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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 Wirtinger's representation and projection theorem ： ウィキペディア英語版 Wirtinger's representation and projection theorem In mathematics, Wirtinger's representation and projection theorem is a theorem proved by Wilhelm Wirtinger in 1932 in connection with some problems of approximation theory. This theorem gives the representation formula for the holomorphic subspace $\backslash left.\backslash right.\; H\_2$ of the simple, unweighted holomorphic Hilbert space $\backslash left.\backslash right.\; L^2$ of functions square-integrable over the surface of the unit disc $\backslash left.\backslash right.\backslash$ of the complex plane, along with a form of the orthogonal projection from $\backslash left.\backslash right.\; L^2$ to $\backslash left.\backslash right.\; H\_2$. Wirtinger's paper contains the following theorem presented also in Joseph L. Walsh's well-known monograph (p. 150) with a different proof. ''If'' $\backslash left.\backslash right.\backslash left.\; F(z)\backslash right.$ ''is of the class'' $\backslash left.\backslash right.\; L^2$ on , ''i.e. : ''where $\backslash left.\backslash right.\; dS$ is the area element, then the unique function $\backslash left.\backslash right.\; f(z)$ of the holomorphic subclass $H\_2\backslash subset\; L^2$, such that'' : $\backslash iint\_|F(z)-f(z)|^2\; \backslash ,\; dS$ ''is least, is given by : The last formula gives a form for the orthogonal projection from $\backslash left.\backslash right.\; L^2$ to $\backslash left.\backslash right.\; H\_2$. Besides, replacement of $\backslash left.\backslash right.\; F(\backslash zeta)$ by $\backslash left.\backslash right.\; f(\backslash zeta)$ makes it Wirtinger's representation for all $f(z)\backslash in\; H\_2$. This is an analog of the well-known Cauchy integral formula with the square of the Cauchy kernel. Later, after the 1950s, a degree of the Cauchy kernel was called reproducing kernel, and the notation $\backslash left.\backslash right.\; A^2\_0$ became common for the class $\backslash left.\backslash right.\; H\_2$. In 1948 Mkhitar Djrbashian extended Wirtinger's representation and projection to the wider, weighted Hilbert spaces $\backslash left.\backslash right.\; A^2\_\backslash alpha$ of functions $\backslash left.\backslash right.\; f(z)$ holomorphic in , which satisfy the condition : and also to some Hilbert spaces of entire functions. The extensions of these results to some weighted $\backslash left.\backslash right.\; A^2\_\backslash omega$ spaces of functions holomorphic in and similar spaces of entire functions, the unions of which respectively coincide with ''all'' functions holomorphic in and the ''whole'' set of entire functions can be seen in. ==See also== * 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「Wirtinger's representation and projection theorem」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.