Polynomial perturbations of hermitian linear functionals and difference equations

TL;DR

This paper introduces a new method for analyzing polynomial modifications of hermitian functionals on the unit circle, characterizing quasi-definiteness via difference equations for Schur parameters, with applications to orthogonal polynomials.

Contribution

It presents a novel approach to study polynomial modifications of arbitrary degree for hermitian functionals, linking quasi-definiteness to recurrence relations of Schur parameters.

Findings

Characterization of quasi-definiteness through difference equations.

Analysis of degree one polynomial perturbations of Lebesgue measure.

Classification of orthogonal polynomials related by linear relations.

Abstract

This paper is devoted to the study of general (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial modifications of arbitrary degree. The main objective is the characterization of the quasi-definiteness of the functionals involved in the problem in terms of a difference equation relating the corresponding Schur parameters. The results are presented in the general framework of (non necessarily quasi-definite) hermitian functionals, so that the maximum number of orthogonal polynomials is characterized by the number of consistent steps of an algorithm based on the referred recurrence for the Schur parameters. Some concrete applications to the study of orthogonal polynomials on the unit circle show the effectiveness of this new approach: an exhaustive and instructive analysis of the functionals coming from a general inverse polynomial perturbation of degree one for the Lebesgue measure; the classification of those pairs of orthogonal polynomials connected by a kind of linear relation with constant polynomial coefficients; and the determination of those orthogonal polynomials whose associated ones are related to a degree one polynomial modification of the original orthogonality functional.