Two Variable Orthogonal Polynomials and Fejér-Riesz Factorization
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We consider bivariate polynomials orthogonal on the bicircle with respect to a positive nondegenerate measure. The theory of scalar and matrix orthogonal polynomials is reviewed with an eye toward applying it to the bivariate case. The lexicographical and reverse lexicographical orderings are used to order the monomials for the Gram–Schmidt procedues and 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 measure. 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.
KeywordsOrthogonal polynomials on the bicircle Fejér-Riesz factorization Recurrence relations Matrix orthogonal polynomials on the unit circle Bivariate orthogonal polynomials Matrix orthogonal polynomials moment problem Recurrence coefficients
AMS Classification Numbers42C05 42C10
I would like to thank Plamen Iliev for suggestions and corrections to this paper. I would also like to thank an anonymous reviewer for very carefully reading and commenting on these lectures.
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