# 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 (2014)

- Volume: 016, Issue: 9, page 1849-1880
- ISSN: 1435-9855

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topGeronimo, 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 -

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