\(N\)-extremal matrices of measures for an indeterminate matrix moment problem
type: Item
instance of: scholarly article
title: \(N\)-extremal matrices of measures for an indeterminate matrix moment problem
zbMATH Open document ID: 0983.42016
author: A. J. Durán
author: Pedro Lopez-Rodriguez
published in: Journal of Functional Analysis
publication date: 2002-02-06T00:00:00Z
review text: Given \(\mu =(\mu_{i,j})_{i,j=1}^N\), a positive definite matrix of measures on \(\mathbb R\) with finite matrix moments \(S_k=\int_{\mathbb R} t^k d\mu \), \(k \geq 0\), denote by \(V_\mu \) the set of positive definite matrices of measures on \(\mathbb R\) having the same \(S_k\), \(k \geq 0\). The (matrix) moment problem is determinate if \(V_\mu =\{\mu \}\), and indeterminate otherwise. An indeterminate moment problem can have the degree of indetermination varying from 1 to \(N\), when it is called completely indeterminate. In analogy with the case \(N=1\), studied by M.\ Riesz, and for a completely indeterminate moment problem, the authors consider the \(N\)-extremal matrices \(W \in V_\mu \), i.e., such that polynomials are dense in \(L^2(W)\). Several properties of \(W\)'s are established, showing similarities but also differences with the classical scalar case (\(N=1\)).
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 1496043
zbMATH Keywords: matrix orthogonal polynomials
zbMATH Keywords: moment problem
zbMATH Keywords: extremal measure
reviewed by: Andrei Martínez Finkelshtein
MaRDI profile type: Publication
full work available at URL: jfan.2000.3585
OpenAlex ID: W2053955148
cites work: Q5508534
cites work: Q3278549
cites work: THE SCATTERING PROBLEM FOR A DISCRETE STURM-LIOUVILLE OPERATOR
cites work: Q5539254
cites work: A generalization of Favard's theorem for polynomials satisfying a recurrence relation
cites work: On Orthogonal Polynomials With Respect to a Positive Definite Matrix of Measures
cites work: Markov's Theorem for Orthogonal Matrix Polynomials
cites work: Orthogonal matrix polynomials: zeros and Blumenthal's theorem
cites work: The \({\mathcal L}^ p\) space of a positive definite matrix of measures and density of matrix polynomials in \({\mathcal L}^ 1\)
cites work: Density Questions for the Truncated Matrix Moment Problem
cites work: Orthogonal matrix polynomials and higher-order recurrence relations
cites work: Q3939931
cites work: Fundamental aspects of the representation theory of Hermitian operators with deficiency index (𝑚,𝑚)
cites work: Riesz's theorem for orthogonal matrix polynomials
cites work: The Nevanlinna parametrization for a matrix moment problem
cites work: Q3334479
DOI: 10.1006/JFAN.2000.3585
description: scientific article; zbMATH DE number 1496043