Two variable orthogonal polynomials on the bicircle and structured matrices

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Title: Two variable orthogonal polynomials on the bicircle and structured matrices
Author(s): Geronimo, Jeffrey S.; Woerdeman, Hugo J. (Hugo Jan), 1962-
Keywords: Bivariate Orthogonal Polynomials; Positive Definite Linear Functionals; Moment Problem; Doubly Toeplitz Matrices; Recurrence Coefficients
Date: 2007
Abstract: 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 Fej´er–Riesz factorization are also given.
Citation: Review of Scientific Instruments, 78(11): pp. 796–825.
URI: http://dx.doi.org/10.1137/060662472; http://hdl.handle.net/1860/2700
In Collections: Drexel Research

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