Journal of the European Mathematical Society


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Volume 16, Issue 9, 2014, pp. 1849–1880
DOI: 10.4171/JEMS/477

Published online: 2014-10-22

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

Jeffrey S. Geronimo[1] and Plamen Iliev[2]

(1) Georgia Institute of Technology, Atlanta, USA
(2) Georgia Institute of Technology, Atlanta, USA

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.

Keywords: Fejér-Riesz factorizations, bivariate Bernstein-Szegö measures, orthogonal polynomials, spectral theory

Geronimo Jeffrey, Iliev Plamen: Fejér–Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle. J. Eur. Math. Soc. 16 (2014), 1849-1880. doi: 10.4171/JEMS/477