Nevanlinna–Pick interpolation について

.
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:\mathbb\to\mathbb

, for all

\lambda_1, \lambda_2 \in \mathbb

,
:

\left|\frac\right| \leq \left|\frac\right|.

Setting

f(\lambda_i)=z_i

, this inequality is equivalent to the statement that the matrix given by
:

\begin \frac & \frac\\ \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

\lambda_1, \lambda_2, z_1, z_2 \in \mathbb

, there exists a holomorphic function

\varphi:\mathbb \to \mathbb

such that

\varphi(\lambda_1) = z_1

and

\varphi(\lambda_2)=z_2

if and only if the Pick matrix
:

\left(\frac\right)_ \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