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

Geronimo, Jeffrey S., and Iliev, Plamen. "Fejér–Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle." Journal of the European Mathematical Society 016.9 (2014): 1849-1880. <http://eudml.org/doc/277445>.

@article{Geronimo2014,
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 $|w|\le 1$. The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight $\frac\{1\}\{4\pi ^2Q(\theta ,\varphi )\}$ 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-Szegö measures on the unit circle.},
author = {Geronimo, Jeffrey S., Iliev, Plamen},
journal = {Journal of the European Mathematical Society},
keywords = {Fejér-Riesz factorizations; bivariate Bernstein-Szegö measures; orthogonal polynomials; spectral theory; Fejér-Riesz factorizations; bivariate Bernstein-Szegö measures; orthogonal polynomials; spectral theory},
language = {eng},
number = {9},
pages = {1849-1880},
publisher = {European Mathematical Society Publishing House},
title = {Fejér–Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle},
url = {http://eudml.org/doc/277445},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Geronimo, Jeffrey S.
AU - Iliev, Plamen
TI - Fejér–Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 9
SP - 1849
EP - 1880
AB - 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 $|w|\le 1$. The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight $\frac{1}{4\pi ^2Q(\theta ,\varphi )}$ 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-Szegö measures on the unit circle.
LA - eng
KW - Fejér-Riesz factorizations; bivariate Bernstein-Szegö measures; orthogonal polynomials; spectral theory; Fejér-Riesz factorizations; bivariate Bernstein-Szegö measures; orthogonal polynomials; spectral theory
UR - http://eudml.org/doc/277445
ER -