Extension of the theorems of Carathéodory-Toeplitz-Schur and Pick
type: Item
instance of: scholarly article
title: Extension of the theorems of Carathéodory-Toeplitz-Schur and Pick
zbMATH Open document ID: 0631.30033
author: Sechiko Takahashi
published in: Pacific Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pjm/1102650156
review text: Let \(z_ 1,...,z_ k\) be distinct points in the unit disc D and let \(c_{i\alpha}\) \((\alpha =0,...,n_ i-1)\) be \(n_ i\) complex numbers for each \(z_ i\). There exists a holomorphic function f in D with \(| f| \leq 1\) whose first \(n_ i\) Taylor coefficients at \(z_ i\) are the prescribed values \(c_{i\alpha}\) if and only if the Hermitian matrix defined by \(z_ i\) and \(c_{i\alpha}\) is positive semidefinite. This result contains both Schur's theorem (1917), derived from the results of Carathéodory and Toeplitz, and Pick's theorem (1915).
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 30E05
zbMATH DE Number: 4025800
zbMATH Keywords: bounded analytic functions
zbMATH Keywords: Carathéodory coefficient problem
MaRDI profile type: Publication
OpenAlex ID: W2023262933
DOI: 10.2140/PJM.1989.138.391
description: scientific article; zbMATH DE number 4025800