JournalsjemsVol. 16, No. 9pp. 1849–1880

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

  • Jeffrey S. Geronimo

    Georgia Institute of Technology, Atlanta, USA
  • Plamen Iliev

    Georgia Institute of Technology, Atlanta, USA
Fejér–Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle cover
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Abstract

We give a complete characterization of the positive trigonometric polynomials Q(θ,φ)Q(\theta,\varphi) on the bi-circle, which can be factored as Q(θ,φ)=p(eiθ,eiφ)2Q(\theta,\varphi)=|p(e^{i\theta},e^{i\varphi})|^2 where p(z,w)p(z, w) is a polynomial nonzero for z=1|z|=1 and w1|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 14π2Q(θ,φ)\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 zz in lexicographical ordering and multiplication by ww 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.

Cite this article

Jeffrey S. Geronimo, Plamen Iliev, Fejér–Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle. J. Eur. Math. Soc. 16 (2014), no. 9, pp. 1849–1880

DOI 10.4171/JEMS/477