Riesz's theorem for orthogonal matrix polynomials
type: Item
instance of: scholarly article
title: Riesz's theorem for orthogonal matrix polynomials
zbMATH Open document ID: 0924.42019
published in: Constructive Approximation
publication date: 1999-11-08T00:00:00Z
full work available at URL: https://idus.us.es/xmlui/handle/11441/41820
review text: This paper extends to the matrix setting the classical results of Riesz and Nevanlinna connecting orthogonal polynomials, extremal solutions of a moment problem and the denseness of the set of polynomials in an \(L^2\) space. Namely, given a positive definite \(N \times N\) matrix of measures \(\nu\) and the corresponding sequence \(\{P_n\}\) of matrix polynomials, orthonormal with respect to \(\nu\), an infinite block Jacobi matrix \(J\) is constructed, whose entries are the coefficients of the three term recurrence relation satisfied by \(\{P_n\}\). The author considers only the completely indeterminate case, that is when both deficiency indices of \(J\) are \(N\). In this setting he defines an analogue of the Stieltjes (or Cauchy) transform of a matrix of measures and describes the image by this transform of all the solutions of the moment problem corresponding to \(\nu\). In doing so, a well-known theorem of Nevanlinna is extended. Finally, Riesz's theorem relating extremality of a measure \(\mu\) solving a moment problem with the denseness of the sequence of orthonormal polynomials in \(L^2(\mu)\) is generalized to the matrix framework, again under the hypothesis that the matrix moment problem is completely indeterminate.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 44A60
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 42C30
zbMATH DE Number: 1271887
zbMATH Keywords: moment problem
zbMATH Keywords: determinacy
zbMATH Keywords: matrix of measures
zbMATH Keywords: matrix orthogonal polynomials
zbMATH Keywords: extremal solutions
author: Pedro Lopez-Rodriguez
reviewed by: Andrei Martínez Finkelshtein
MaRDI profile type: Publication
OpenAlex ID: W2162849908
Wikidata QID: Q127359279
DOI: 10.1007/S003659900101
description: scientific article; zbMATH DE number 1271887