Two variable orthogonal polynomials on the bicircle and structured matrices

Review of Scientific Instruments, 78(11): pp. 796–825.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

Fej\'er-Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle

We give a complete characterization of the positive trigonometric polynomials Q(\theta,\phi) on the bi-circle, which can be factored as Q(\theta,\phi)=|p(e^{i\theta},e^{i\phi})|^2 where p(z,w) is a polynomial nonzero for |z|=1 and |w|\leq 1. The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight 1/(4\pi^2Q(\theta,\phi)) on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities relating the recurrence coefficients for the corresponding polynomials and vice versa. In particular, we characterize the Borel measures on the bi-circle for which the coefficients multiplying the reverse polynomials associated with the two operators: multiplication by z in lexicographical ordering and multiplication by w in reverse lexicographical ordering vanish after a particular point. This can be considered as a spectral type result analogous to the characterization of the Bernstein-Szeg\H{o} measures on the unit circle

Orthogonality relations for bivariate Bernstein-Szeg\H{o} measures

The orthogonality properties of certain subspaces associated with bivariate Bernstein-Szeg\H{o} measures are considered. It is shown that these spaces satisfy more orthogonality relations than expected from the relations that define them. The results are used to prove a Christoffel-Darboux like formula for these measures.Comment: 15 pages, 3 figure

Polynomials with no zeros on a face of the bidisk

We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement, namely no zeros on a face of the bidisk. Two different characterizations are given using a Hilbert space structure naturally associated to the trigonometric polynomial; one is in terms of a certain orthogonal decomposition the Hilbert space must possess called the “split-shift orthogonality condition” and another is an operator theoretic or matrix condition closely related to an earlier characterization due to the first two authors. This approach allows several refinements of the characterization and it also allows us to prove a sums of squares decomposition which at once generalizes the Cole–Wermer sums of squares result for two variable stable polynomials as well as a sums of squares result related to the Schur–Cohn method for counting the roots of a univariate polynomial in the unit disk

Tridiagonal Operators and Zeros of Polynomials in Two Variables

The aim of this paper is to connect the zeros of polynomials in two variables with the eigenvalues of a self-adjoint operator. This is done by use of a functional-analytic method. The polynomials in two variables are assumed to satisfy a five-term recurrence relation, similar to the three-term recurrence relation that the classical orthogonal polynomials satisfy

The Analytic Theory of Matrix Orthogonal Polynomials

We give a survey of the analytic theory of matrix orthogonal polynomials.Comment: 85 page