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

Abstract

We give a complete characterization of the positive trigonometric polynomials Q(\theta,\varphi) on the bi-circle, which can be factored as Q(\theta,\varphi)=|p(e^{i\theta},e^{i\varphi})|^2 where p(z, w) is a polynomial nonzero for |z|=1 and |w|\leq1 . The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight \frac{1}{4\pi^2Q(\theta,\varphi)} 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.

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• Journal of the European Mathematical Society - View - PDF