Two Variable Orthogonal Polynomials on the Bi-circle and Structured Matrices

TL;DR

This paper develops recurrence relations for bivariate orthogonal polynomials on the bicircle, linking different orderings, and explores their applications to positive measures and Fejér-Riesz factorization.

Contribution

It introduces new recurrence formulas connecting lexicographical and reverse lexicographical orthogonal polynomials on the bicircle, and characterizes conditions for positive linear functionals.

Findings

Derived recurrence relations between polynomial orderings

Established conditions for existence of positive measures

Applied results to Fejér-Riesz factorization

Abstract

We consider bivariate polynomials orthogonal on the bicircle with respect to a positive linear functional. The lexicographical and reverse lexicographical orderings are used to order the monomials. Recurrence formulas are derived between the polynomials of different degrees. These formulas link the orthogonal polynomials constructed using the lexicographical ordering with those constructed using the reverse lexicographical ordering. Relations between the coefficients in the recurrence formulas are derived and used to give necessary and sufficient conditions for the existence of a positive linear functional. These results are then used to construct a class of two variable measures supported on the bicircle that are given by one over the magnitude squared of a stable polynomial. Applications to Fej\'er-Riesz factorization are also given