Dual Szegő pairs of sequences of rational matrix-valued functions
type: Item
instance of: scholarly article
title: Dual Szegő pairs of sequences of rational matrix-valued functions
zbMATH Open document ID: 1121.42016
DOI: 10.1155/IJMMS/2006/23723
author: Andreas Lasarow
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2007-06-13T00:00:00Z
full work available at URL: https://eudml.org/doc/53455
review text: This paper is the dual of [\textit{B. Fritzsche, B. Kirstein} and \textit{A. Lasarow}, ``Szegő pairs of orthogonal rational matrix-valued functions on the unit circle''. in Operator theory and indefinite inner product spaces. Basel: Birkhäuser. Operator Theory: Adv. Appl. 163, 163--189 (2006; Zbl 1120.42014)]. Matrix-valued rational functions with prescribed poles, orthogonal with respect to a matrix-valued measure supported on the complex unit circle satisfy a Szegő type of recurrence relation, characterized by matrix-valued Szegő parameters. In these relations, left and right matrix-valued orthogonal rational functions (MORF) are mixed, hence they form Szegő pairs. Changing the sign of the Szegő parameters gives a dual Szegő pair. These dual pairs also satisfy some Christoffel-Darboux relations and they turn out to be MORF for a (matrix-valued) measure that is closely related to the orthogonality measure of the MORF. Also the relation between the recurrence relation and Potapov's J-theory is given.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 47A56
zbMATH DE Number: 5164012
zbMATH Keywords: matrix valued functions
zbMATH Keywords: Christoffel-Darboux relation
zbMATH Keywords: orthogonal rational functions
zbMATH Keywords: Szegő parameters
reviewed by: Adhemar Bultheel
MaRDI profile type: Publication
cites work: Discrete and continuous boundary problems
cites work: Matrix-valued nevanlinna-pick interpolation with complexity constraint: an optimization approach
cites work: Q4227851
cites work: Q4227852
cites work: Q4223056
cites work: ORTHOGONAL RATIONAL FUNCTIONS AND INTERPOLATORY PRODUCT RULES ON THE UNIT CIRCLE
cites work: On the multiple Nevanlinna-Pick matrix interpolation in the class \(\mathcal C_p\) and the Carathéodory matrix coefficient problem
cites work: Orthogonal polynomial matrices on the unit circle
cites work: Schur Parametrization of Positive Definite Block-Toeplitz Systems
cites work: Q3995393
cites work: Q3032797
cites work: Q5332729
cites work: Q5685941
cites work: The commutant lifting approach to interpolation problems
cites work: An inverse entropy optimization problem for matrix-valued carathéodory functions
cites work: An Extension Problem for Non-Negative Hermitian Block Toeplitz Matrices
cites work: An Extension Problem for Non-negative Hermitian Block Toeplitz Matrices. II
cites work: An Extension Problem for Non-Negative Hermitian Block Toeplitz Matrices. III
cites work: Q3810043
cites work: An Extension Problem for Non-Negative Hermitian Block Toeplitz Matrices, IV
cites work: An Extension Problem for Non-Negative Hermitian Block Toeplitz Matrices, V
cites work: On rank invariance of moment matrices of nonnegative Hermitian-valued Borel measures on the unit circle
cites work: On rank invariance of Schwarz-Pick-Potapov block matrices of matricial Schur functions
cites work: Orthogonal rational matrix‐valued functions on the unit circle
cites work: Q5477799
cites work: Orthogonal rational matrix‐valued functions on the unit circle: Recurrence relations and a Favard‐type theorem
cites work: Q3795535
cites work: Q5835454
cites work: On the trigonometric moment problem
cites work: Orthogonal polynomials
cites work: Q3247378
cites work: Solution of a multiple Nevanlinna--Pick problem via orthogonal rational functions
cites work: Q3794320
cites work: The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure
cites work: Orthogonal matrix polynomials and applications
cites work: Q5777281
cites work: Bauer-type factorization of positive matrices and the theory of matrix polynomials orthogonal on the unit circle
description: scientific article; zbMATH DE number 5164012