Dual Szegö pairs of sequences of rational matrix‐valued functions

Andreas Lasarow

doi:10.1155/ijmms/2006/23723

Abstract

We study certain sequences of rational matrix‐valued functions with poles outside the unit circle. These sequences are recursively constructed based on a sequence of complex numbers with norm less than one and a sequence of strictly contractive matrices. We present some basic facts on the rational matrix‐valued functions belonging to such kind of sequences and we will see that the validity of some Christoffel‐Darboux formulae is an essential property. Furthermore, we point out that the considered dual pairs consist of orthogonal systems. In fact, we get similar results as in the classical theory of Szegö′s orthogonal polynomials on the unit circle of the first and second kind.

Highlights

The theory of orthogonal polynomials is known to have numerous applications in an extensive range of engineering problems
Starting from different points of view of applications Bultheel, Gonzalez-Vera, Hendriksen, and Njastad have formed up a fruitful collaboration and created in the 1990s a comprehensive theory of scalar orthogonal rational functions on the unit circle
In a series of research papers they worked systematically out basic parts of a concept of generalizing essential parts of the classical theory of orthogonal polynomials on the unit circle

Summary

Introduction

The theory of orthogonal polynomials is known to have numerous applications in an extensive range of engineering problems. In a forthcoming work, these formulae will play a key role by solving interpolation problems of Nevanlinna-Pick type for matrix-valued Caratheodory functions in D via orthogonal rational matrix-valued functions including an interrelation between the parameters which appear in the recurrence relations studied in the present paper and the parameters which appear in the algorithm discussed in [24, Section 5]. Fact, [26, Theorems 2.11, 3.5, and 3.7] imply a parametrization of these particular pairs [(Xj)τj=0, (Yj)τj=0] of orthogonal rational matrix-valued functions in terms of an initial condition and a sequence (E )τ=1 of strictly contractive q × q matrices These considerations are the starting point for the present paper. Szego’s classical orthogonal polynomials of the first and the second kind

Some basic facts

Connection to Potapov’s J-theory

Christoffel-Darboux formulae

A characterization of Szegopairs

On particular measures corresponding to the dual pairs