On Hankel nonnegative definite sequences, the canonical Hankel parametrization, and orthogonal matrix polynomials
type: Item
instance of: scholarly article
title: On Hankel nonnegative definite sequences, the canonical Hankel parametrization, and orthogonal matrix polynomials
zbMATH Open document ID: 1298.05016
author: B. Fritzsche
author: B. Kirstein
author: Conrad Mädler
published in: Complex Analysis and Operator Theory
publication date: 2011-11-04T00:00:00Z
review text: The authors study several classes of finite sequences of complex \(q\times q\) matrices, in connection with two matricial versions of the truncated Hamburger moment problem. Specifically, these classes of finite sequences of matrices are related to the following matrix versions of the truncated moment problem:{\parindent=8mm \begin{itemize}\item[(i)] Let \(n\in\mathbb N_0\) and let \((s_j)_{j=0}^{2n}\) be a sequence of complex \(q\times q\) matrices. One should describe the class of all nonnegative Hermitian \(q\times q\) Borel measures \(\sigma\) on \(\mathbb R\) such that for each nonnegative integer \(j\leq2n\) the integral \(s_j^{|\sigma|}=\int_{\mathbb R}t^j\sigma(dt)\) exists, where \(s_k^{|\sigma|}=s_k\) if \(0\leq k\leq 2n-1\) and \(s_{2N}^{|\sigma|}-s_{2n}\) is nonnegative Hermitian. \item[(ii)] The second problem is a version of (i), with \(2n\) replaced by a not necessarily even integer \(m\), such that \(s_k^{|\sigma|}=s_k\) if \(0\leq k\leq m\). \end{itemize}} These moment problems were previously studied by \textit{V. Bolotnikov} [Integral Equations Oper. Theory 25, No. 3, 253--276 (1996; Zbl 0856.47014)], \textit{G. Chen} and \textit{Y. Hu} [Linear Algebra Appl. 277, No. 1--3, 199--236 (1998; Zbl 0932.44005)], and also by \textit{Y. M. Dyukarev} et al. [Complex Anal. Oper. Theory 3, No. 4, 759--834 (2009; Zbl 1186.44006)] via a canonical Hankel parametrization. In the present paper, the authors investigate further aspects of this canonical Hankel parametrization.
reviewed by: Florian-Horia Vasilescu
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 47A57
Mathematics Subject Classification ID: 44A60
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 15B10
zbMATH DE Number: 5968091
zbMATH Keywords: nonnegative Hermitian measures
zbMATH Keywords: truncated matricial Hamburger moment problem
zbMATH Keywords: nonnegative Hermitian block Hankel matrices
zbMATH Keywords: Hankel nonnegative definite sequences
zbMATH Keywords: Hankel nonnegative definite extendable sequences
zbMATH Keywords: canonical Hankel parametrization
zbMATH Keywords: orthogonal matrix polynomials
zbMATH Keywords: sequences of moments of classical probability distributions
zbMATH Keywords: generalized Lucas numbers
zbMATH Keywords: Catalan parametrization
describes a project that uses: OEIS
MaRDI profile type: Publication
full work available at URL: s11785-010-0054-9
OpenAlex ID: W2051261265
cites work: A Course in Enumeration
cites work: Q3550093
cites work: Generalized inverses. Theory and applications.
cites work: On degenerate Hamburger moment problem and extensions of nonnegative Hankel block matrices
cites work: The truncated Hamburger matrix moment problems in the nondegenerate and degenerate cases, and matrix continued fractions
cites work: Q4483926
cites work: Q3995393
cites work: On distinguished solutions of truncated matricial Hamburger moment problems
cites work: Q5387515
cites work: Q3795535
cites work: Scattering theory and matrix orthogonal polynomials on the real line
cites work: Hankel determinants and orthogonal polynomials.
cites work: The Hankel transform and some of its properties
cites work: Matrix-valued orthogonal polynomials on the real line: some extensions of the classical theory
cites work: The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure
cites work: Operators as spectral integrals of operator-valued functions from the study of multivariate stationary stochastic processes
cites work: Spectral integrals of operator-valued functions. II: From the study of stationary processes
cites work: On a generalization of the classical moment problem
cites work: The On-Line Encyclopedia of Integer Sequences
cites work: Q5395429
cites work: Some aspects of Hankel matrices in coding theory and combinatorics
DOI: 10.1007/S11785-010-0054-9
description: scientific article; zbMATH DE number 5968091