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Two Variable Orthogonal Polynomials and Fejér-Riesz Factorization

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Abstract

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.

Based on lectures given for the AIMS-Volkswagen Foundation Workshop on Introduction to Orthogonal Polynomials and Applications.

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Acknowledgements

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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Authors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, USA

    J. S. Geronimo

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  1. J. S. Geronimo
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Correspondence to J. S. Geronimo .

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Editors and Affiliations

  1. University of Yaoundé I, Yaoundé, Cameroon; African Institute for Mathematical Sciences, Limbe, Cameroon

    Mama Foupouagnigni

  2. Institute for Mathematics, University of Kassel, Kassel, Germany

    Wolfram Koepf

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Geronimo, J.S. (2020). Two Variable Orthogonal Polynomials and Fejér-Riesz Factorization. In: Foupouagnigni, M., Koepf, W. (eds) Orthogonal Polynomials. AIMSVSW 2018. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36744-2_14

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