Fejér–Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle

Jeffrey S. Geronimo   ; Plamen Iliev

Journal of the European Mathematical Society, Tome 16 (2014) no. 9, pp. 1849-1880

Résumé

We give a complete characterization of the positive trigonometric polynomials Q(θ,φ) on the bi-circle, which can be factored as Q(θ,φ)=∣p(eiθ,eiφ)∣2 where p(z,w) is a polynomial nonzero for ∣z∣=1 and ∣w∣≤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 4π2Q(θ,φ)1​ 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.

Détail

DOI : 10.4171/jems/477

Classification : 42-XX, 00-XX, 30-XX, 47-XX
Keywords: Fejér-Riesz factorizations, bivariate Bernstein-Szegö measures, orthogonal polynomials, spectral theory