2006/12/14 by Jeffrey S. Geronimo, Geronimo, Jeffrey S., Hugo J. Woerdeman +2
Computer Science · Mathematics · Physics and Astronomy · #30E05 #42C05 #47A57 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #Scientific Research and Discoveries #Spectral Theory (math.SP) #math.CA #math.SP #msc:30E05 #msc:42C05 #msc:47A57
paper · pdf · doi:10.48550/arxiv.math/0612400
arxiv created 2006/12/14 · openalex publication_date 2006/12/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider bivariate polynomials orthogonal on the bicircle with respect to a positive linear functional. The lexicographical and reverse lexicographical orderings are used to order the monomials. Recurrence formulas are derived between the polynomials of different degrees. These formulas link the orthogonal polynomials constructed using the lexicographical ordering with those constructed using the reverse lexicographical ordering. Relations between the coefficients in the recurrence formulas are derived and used to give necessary and sufficient conditions for the existence of a positive linear functional. These results are then used to construct a class of two variable measures supported on the bicircle that are given by one over the magnitude squared of a stable polynomial. Applications to Fejér-Riesz factorization are also given