Geodesic
Fejér–Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle
Journal of the European Mathematical Society, Tome 16 (2014) no. 9, pp. 1849-1880.

Voir la notice de l'article provenant de la source EMS Press

We give a complete characterization of the positive trigonometric polynomials Q(θ,φ) on the bi-circle, which can be factored as Q(θ,φ)=∣p(eiθ,eiφ)∣2 where p(z,w) is a polynomial nonzero for ∣z∣=1 and ∣w∣≤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 4π2Q(θ,φ)1​ 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.
DOI : 10.4171/jems/477
Classification : 42-XX, 00-XX, 30-XX, 47-XX
Keywords: Fejér-Riesz factorizations, bivariate Bernstein-Szegö measures, orthogonal polynomials, spectral theory 
@article{JEMS_2014_16_9_a2,
     author = {Jeffrey S. Geronimo and Plamen Iliev},
     title = {Fej\'er{\textendash}Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle},
     journal = {Journal of the European Mathematical Society},
     pages = {1849--1880},
     publisher = {mathdoc},
     volume = {16},
     number = {9},
     year = {2014},
     doi = {10.4171/jems/477},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/477/}
}
TY  - JOUR
AU  - Jeffrey S. Geronimo
AU  - Plamen Iliev
TI  - Fejér–Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle
JO  - Journal of the European Mathematical Society
PY  - 2014
SP  - 1849
EP  - 1880
VL  - 16
IS  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.4171/jems/477/
DO  - 10.4171/jems/477
ID  - JEMS_2014_16_9_a2
ER  - 
%0 Journal Article
%A Jeffrey S. Geronimo
%A Plamen Iliev
%T Fejér–Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle
%J Journal of the European Mathematical Society
%D 2014
%P 1849-1880
%V 16
%N 9
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.4171/jems/477/
%R 10.4171/jems/477
%F JEMS_2014_16_9_a2
Jeffrey S. Geronimo; Plamen Iliev. Fejér–Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle. Journal of the European Mathematical Society, Tome 16 (2014) no. 9, pp. 1849-1880. doi : 10.4171/jems/477. http://geodesic.mathdoc.fr/articles/10.4171/jems/477/

Cité par Sources :