The Nevanlinna-Pick interpolation problems and power moment problems for matrix-valued functions. III: The infinitely many data case
type: Item
instance of: scholarly article
title: The Nevanlinna-Pick interpolation problems and power moment problems for matrix-valued functions. III: The infinitely many data case
zbMATH Open document ID: 0953.47013
DOI: 10.1016/S0024-3795(99)00241-4
published in: Linear Algebra and its Applications
publication date: 2000-08-07T00:00:00Z
review text: The paper deals with the multiple Nevanlinna-Pick (NP) matrix interpolation problem with a denumerable set of nodes in the Nevanlinna class \(\mathcal{N}_p\) \((p \geq 1)\) of \( \mathbb{C}^{p \times p}\)-valued functions \(f(z)\) which are holomorphic in the open upper half plane \(\pi^+\) (NP\((\mathcal{N}_p)_\infty\) problem) and the full Hamburger matrix moment problem (HM\((\mathbb{R})_\infty\) problem). The close connection between an NP\((\mathcal{N}_p)_\infty\) problem and HM\((\mathbb{R})_\infty\) problem associated with the block Hankel vector of the former is studied. This connection allows to reduce the solution of the NP\((\mathcal{N}_p)_\infty\) problem to the study of a certain HM\((\mathbb{R})_\infty\) problem. The case of infinitely many nodes is handled by means of analysis of the limiting behaviour of approximating problems involving only finitely many nodes. The notion of the block Hankel vector of an NP\((\mathcal{N}_p)_\infty\) problem is first introduced and an integral and explicit correspondence between solutions to a solvable NP\((\mathcal{N}_p)_\infty\) problem and solutions to the associated HM\((\mathbb{R})_\infty\) problem with that block Hankel vector is established. Then a congruence relation between the infinite generalized block Loewner matrix \(L_\infty\) of NP\((\mathcal{N}_p)_\infty\) problem and the infinite block Hankel matrix \(H_\infty\) built on the block Hankel vector of the former problem is made, which is a generalization of the classical Loewner-Hankel matrix relation. This link allows to obtain a parametrization of the solutions to the multiple NP\((\mathcal{N}_p)_\infty\) problem in both nondegenerate and degenerate cases by a linear fractional transformation from the corresponding results of the HM\((\mathbb{R})_\infty\) problem. The transformation is based on the Schur algorithm involving matrix continued fractions as well as orthogonal polynomial matrices simultaneously.
reviewed by: Pavel Dyshlovenko
Mathematics Subject Classification ID: 47A57
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 1421385
zbMATH Keywords: Nevanlinna-Pick interpolation
zbMATH Keywords: Hamburger moment problem
zbMATH Keywords: Nevanlinna class
zbMATH Keywords: block Hankel vector
zbMATH Keywords: generalized Schur complement
zbMATH Keywords: Weyl matrix disk
zbMATH Keywords: nonnegatively extendable
zbMATH Keywords: denumerable set of nodes
zbMATH Keywords: infinite generalized block Loewner matrix
zbMATH Keywords: Loewner-Hankel matrix relation
zbMATH Keywords: linear fractional transformation
zbMATH Keywords: Schur algorithm
zbMATH Keywords: matrix continued fractions
zbMATH Keywords: orthogonal polynomial matrices
author: Gongning Chen
author: Yongjian Hu
MaRDI profile type: Publication
cites work: Q5508534
cites work: Q4081840
cites work: Q5601153
cites work: Q3977380
cites work: Errata to: Two-sided tangential interpolation of real rational matrix functions
cites work: Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: Parametrization of the set of all solutions
cites work: On degenerate Hamburger moment problem and extensions of nonnegative Hankel block matrices
cites work: The general rational interpolation problem and its connection with the Nevanlinna-Pick interpolation and power moment problem
cites work: The truncated Hamburger matrix moment problems in the nondegenerate and degenerate cases, and matrix continued fractions
cites work: On the multiple Nevanlinna-Pick matrix interpolation in the class \(\mathcal C_p\) and the Carathéodory matrix coefficient problem
cites work: The Nevanlinna-Pick interpolation problems and power moment problems for matrix-valued functions
cites work: Q5463720
cites work: More on Loewner matrices
cites work: The Nevanlinna–Pick Problem for Matrix-Valued Functions
cites work: Schur Parametrization of Positive Definite Block-Toeplitz Systems
cites work: Q3032797
cites work: On hermitian block Hankel matrices, matrix polynomials, the Hamburger moment problem, interpolation and maximum entropy
cites work: Multiplicative and additive classes of Stieltjes analytic matrix-valued functions, and interpolation problems associated with them. I
cites work: Q4052752
cites work: Q4097968
cites work: The commutant lifting approach to interpolation problems
cites work: Q4317788
cites work: Q5185485
cites work: Hamburger moment problem: Parseval equality and \(A\)-singularity
cites work: ANALYTIC THEORY OF A CLASS OF INTERPOLATION PROBLEMS
cites work: Q3961739
cites work: An indefinite metric in the Nevanlinna-Pick problem
cites work: Q4134389
cites work: Q4163734
cites work: An interpolation problem in the class of Stieltjes functions and its connection with other problems
cites work: Q4120977
cites work: Q4284370
cites work: The multiplicative structure of 𝐽-contractive matrix functions
cites work: An operator-theoretic approach to theorems of the Pick-Nevanlinna and Loewner types. I
cites work: Nevanlinna-Pick interpolation with boundary data
cites work: Q5845997
cites work: A unified approach to Loewner and Hankel matrices
Wikidata QID: Q127352764
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description: scientific article; zbMATH DE number 1421385