On Hankel nonnegative definite sequences, the canonical Hankel parametrization, and orthogonal matrix polynomials
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.
truncated matricial Hamburger moment problem
nonnegative Hermitian block Hankel matrices
Hankel nonnegative definite extendable sequences
sequences of moments of classical probability distributions
scientific article; zbMATH DE number 5968091
scientific article