Two Variable Orthogonal Polynomials and Fejér-Riesz Factorization

  • J. S. GeronimoEmail author
Conference paper
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)


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.


Orthogonal 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 Numbers

42C05 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.


  1. 1.
    M. Bakonyi, H.J. Woerdeman, Matrix Completions, Moments, and Sums of Hermitian Squares (Princeton University Press, Princeton, 2011)CrossRefGoogle Scholar
  2. 2.
    J.P. Burg, Maximum entropy spectral analysis. Ph.D. Dissertation, Stanford University, Stanford (1975)Google Scholar
  3. 3.
    D. Damanik, A. Pushnitski, B. Simon, The analytic theory of matrix orthogonal polynomials. Surv. Approx. Theory 4, 1–85 (2008)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ph. Delsarte, Y.V. Genin, Y.G. Kamp, Orthogonal polynomial matrices on the unit circle. IEEE Trans. Circuits Syst. 25(3), 149–160 (1978)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ph. Delsarte, Y.V. Genin, Y.G. Kamp, Planar least squares inverse polynomials. I. Algebraic properties. IEEE Trans. Circuits Syst. 26(1), 59–66 (1979)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ph. Delsarte, Y.V. Genin, Y.G. Kamp, A simple proof of Rudin’s multivariable stability theorem. IEEE Trans. Acoust. Speech Signal Process. 28(6), 701–705 (1980)MathSciNetCrossRefGoogle Scholar
  7. 7.
    J.S. Geronimo, P. Iliev, Fejér-Riesz factorizations and the structure of bivariate polynomials orthogonal on the bicircle. J. Eur. Math. Soc. 16(9), 1849–1880 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    J.S. Geronimo, H.J. Woerdeman, Positive extensions, Fejér-Riesz factorization and autoregressive filters in two variables. Ann. Math. 160(3), 839–906 (2004)zbMATHGoogle Scholar
  9. 9.
    J.S. Geronimo, H.J. Woerdeman, Two variable orthogonal polynomials on the bicircle and structured matrices. SIAM J. Matrix Anal. Appl. 29(3), 796–825 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    J.S. Geronimo, P. Iliev, G. Knese, Orthogonality relations for bivariate Bernstein-Szegő measures, in Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications. Contemporary Mathematics, vol. 578 (American Mathematical Society, Providence, 2012), pp. 119–131Google Scholar
  11. 11.
    J.S. Geronimo, P. Iliev, G. Knese, Polynomials with no zeros on a face of the bidisk. J. Funct. Anal. 270, 3505–3558 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ya. L. Geronimus, Orthogonal Polynomials (Consultants Bureau, New York, 1961)Google Scholar
  13. 13.
    H. Helson, D. Lowdenslager, Prediction theory and Fourier series in several variables. Acta Math. 99, 165–202 (1958); II, Acta Math. 106, 175– 213 (1961)Google Scholar
  14. 14.
    T. Kailath, A. Vieira, M. Morf, Inverses of Toeplitz operators, innovations, and orthogonal polyomials. SIAM Rev. 20, 106–119 (1978)MathSciNetCrossRefGoogle Scholar
  15. 15.
    G. Knese, Bernstein-Szegő measures on the two dimensional torus. Indiana Univ. Math. J. 57(3), 1353–1376 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    G. Knese, Polynomials with no zeros on the bidisk. Anal. PDE 3(2) 109–149 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    H.J. Landau, Z. Landau, On the trigonometric moment problem in two dimensions. Indag. Math. (N.S.) 23 (4), 1118–1128 (2012)Google Scholar
  18. 18.
    B. Simon, Orthogonal polynomials on the unit circle, in Part 1, American Mathematical Society Colloquium Publications, vol. 54 (American Mathematical Society, Providence, 2005)Google Scholar
  19. 19.
    G. Szegő, Orthogonal Polynomials, in American Mathematical Society Colloquium Publications, vol. 23 (American Mathematical Society, Providence, 1939 (revised edition 1959))Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Georgia Institute of TechnologyAtlantaUSA

Personalised recommendations