Two variable deformations of the Chebyshev measure

We construct one and two parameter deformations of the two dimensional Chebyshev polynomials with simple recurrence coefficients, following the algorithm in [3]. Using inverse scattering techniques, we compute the corresponding orthogonality measures.Comment: 16 page

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

Factorization of Multivariate Positive Laurent Polynomials

Recently Dritschel proves that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Fejer-Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. Finally we discuss how to compute nonnegative Laurent polynomial factorizations in the multivariate setting