Matrix moment perturbations and the inverse Szegő matrix transformation

Edinson Fuentes,Luis E Garza

doi:10.33044/revuma.v60n2a19

Abstract

Given a perturbation of a matrix measure supported on the unit circle, we analyze the perturbation obtained on the corresponding matrix measure on the real line, when both measures are related through the Szeg˝o matrix transformation. Moreover, using the connection formulas for the corresponding sequences of matrix orthogonal polynomials, we deduce several properties such as relations between the corresponding norms. We illustrate the obtained results with an example.

Highlights

The term moment problem was used for the first time in T
In the last 30 years, several known properties of orthogonal polynomials in the scalar case have been extended to the matrix case, such as algebraic aspects related to their zeros, recurrence relations, Favard type theorems, and Christoffel–Darboux formulas, among many others
As in the scalar case, matrix orthogonal polynomials have proved to be a useful tool in the analysis of many problems of mathematics such as differential equations [13, 28], rational approximation theory [20], spectral theory of Jacobi matrices [1, 29], analysis of polynomial sequences satisfying higher order recurrence relations [11, 17], quantum random walks [3], and Gaussian quadrature formulas [2, 12, 16, 31], among many others

Summary

Introduction

The term moment problem was used for the first time in T. As in the scalar case, matrix orthogonal polynomials have proved to be a useful tool in the analysis of many problems of mathematics such as differential equations [13, 28], rational approximation theory [20], spectral theory of Jacobi matrices [1, 29], analysis of polynomial sequences satisfying higher order recurrence relations [11, 17], quantum random walks [3], and Gaussian quadrature formulas [2, 12, 16, 31], among many others In this contribution, we are interested in the study of some properties related with a perturbation of a sequence of matrix moments, within the framework of the theory of matrix orthogonal polynomials both on the real line and on the unit circle.

Matrix orthogonal polynomials

Connection formulas on the real line and further results

Connection formulas for the unit circle and further results