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Preprint
Two Variable Orthogonal Polynomials on the Bi-circle and Structured Matrices
ArXiv.org
14 Dec 2006
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
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Details
- Title
- Two Variable Orthogonal Polynomials on the Bi-circle and Structured Matrices
- Creators
- Jeffrey S GeronimoHugo Woerdeman
- Publication Details
- ArXiv.org
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021864940604721