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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 Nevanlinna–Pick interpolation ： ウィキペディア英語版 Nevanlinna–Pick interpolation In complex analysis, given ''initial data'' consisting of $n$ points $\backslash lambda\_1,\; \backslash ldots,\; \backslash lambda\_n$ in the complex unit disc $\backslash mathbb$ and ''target data'' consisting of $n$ points $z\_1,\; \backslash ldots,\; z\_n$ in $\backslash mathbb$, the Nevanlinna–Pick interpolation problem is to find a holomorphic function $\backslash varphi$ that interpolates the data, that is for all $i$, :$\backslash varphi(\backslash lambda\_i)\; =\; z\_i$, subject to the constraint $\backslash left\backslash vert\; \backslash varphi(\backslash lambda)\; \backslash right\backslash vert\; \backslash le\; 1$ for all $\backslash lambda\; \backslash in\; \backslash mathbb$. Georg Pick and Rolf Nevanlinna solved the problem independently in 1916 and 1919 respectively, showing that an interpolating function exists if and only if a matrix defined in terms of the initial and target data is positive semi-definite. ==Background== The Nevanlinna-Pick theorem represents an $n$ point generalization of the Schwarz lemma. The invariant form of the Schwarz lemma states that for a holomorphic function $f:\backslash mathbb\backslash to\backslash mathbb$, for all $\backslash lambda\_1,\; \backslash lambda\_2\; \backslash in\; \backslash mathbb$, :$\backslash left|\backslash frac\backslash right|\; \backslash leq\; \backslash left|\backslash frac\backslash right|.$ Setting $f(\backslash lambda\_i)=z\_i$, this inequality is equivalent to the statement that the matrix given by : \\ \frac & \frac \end \geq 0, that is the ''Pick matrix'' is positive semidefinite. Combined with the Schwarz lemma, this leads to the observation that for $\backslash lambda\_1,\; \backslash lambda\_2,\; z\_1,\; z\_2\; \backslash in\; \backslash mathbb$, there exists a holomorphic function $\backslash varphi:\backslash mathbb\; \backslash to\; \backslash mathbb$ such that $\backslash varphi(\backslash lambda\_1)\; =\; z\_1$ and $\backslash varphi(\backslash lambda\_2)=z\_2$ if and only if the Pick matrix :$\backslash left(\backslash frac\backslash right)\_\; \backslash geq\; 0.$ 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「Nevanlinna–Pick interpolation」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.