Polynomials orthogonal on the unit circle with random recurrence coefficients

  • J. S. Geronimo
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1550)

Abstract

Polynomials orthogonal on the unit circle whose recurrence coefficients are generated from a stationary stochastic process are considered. A Lyapunov exponent introduced and its properties are related to absolutely continuous components of the orthogonality measure.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Akhiezer N. I., The Classical Moment Problem, Hafner, N.Y., 1965.MATHGoogle Scholar
  2. [2]
    Craig W. and Simon B., Subharmonicity of the Lyapunov index, Duke Math. J. 50 (1983), pp. 551–560.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    Deift P. and Simon B., Almost periodic Schrödinger operators III. The absolutely continuous spectrum in one dimension, Commun. Math. Phys. 90 (1983), pp. 389–411.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    Delsarte P. and Genin Y., The tridiagonal approach to Szegö's orthogonal polynomials, Toeplitz systems, and related interpolation problems, SIAM J. Math. Anal. 19 (1988), pp. 718–738.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Geronimo J. S., W., Harrell H. E. M. and Van Assche W., On the asymptotic distribution of eigenvalues of banded matrices, Constr. Approx. 4 (1988), pp. 403–417.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Geronimo J. S. and Johnson R., Rotation numbers associated with difference equations satisfied by polynomials orthogonal on the unit circle, in preparation.Google Scholar
  7. [7]
    Geronimus Yu L., Polynomials Orthogonal on a Circle and Interval, Pergamon Press, NY, 1960.MATHGoogle Scholar
  8. [8]
    Herman M., Une methode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractere local d'un theoreme d'Arnold et de Moser sur le tore de dimension 2, Comment. Math. Helvetici 58 (1982), pp. 453–502.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    Kotani S., Lyapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators, Proc. Taniguchi Symp. SA Katata, (1982), pp. 225–247.Google Scholar
  10. [10]
    Krengel U., Ergodic Theorems, de Gruyter, NY, 1985.CrossRefMATHGoogle Scholar
  11. [11]
    Nikishin E.M., Random Orthogonal Polynomials on a circle, Vestnik. Moskovskogo Universitita Matematika 42 (1987), pp. 52–55.MathSciNetMATHGoogle Scholar
  12. [12]
    Rosenblum M. and Rovnyak J., Hardy Classes and Operator Theory, Clarendon, NY, 1985.MATHGoogle Scholar
  13. [13]
    Ruelle D., Characteristic exponents and invariant manifolds in Hilbert space, Annal of Math. 115 (1982), pp. 243–290.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    Simon B., Kotani theory for one dimensional stochastic Jacobi matrices,, Commun. Math. Phys. 89 (1983), pp. 227–234.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Simon B., Schrödinger semigroups, Bull. AMS 7 (1982), pp. 447–526.CrossRefMATHGoogle Scholar
  16. [16]
    Teplyaev A.V., Orthogonal polynomials with random recurrence coefficients, LOMI Scientific Seminar notes 117 (1989).Google Scholar
  17. [17]
    Szegö G., Orthogonal polynomials, AMS Coll. Pub. 23 (1978).Google Scholar

Copyright information

© The Euler International Mathematical Institute 1993

Authors and Affiliations

  • J. S. Geronimo
    • 1
  1. 1.Math. Dept.Georgia Tech.AtlantaUSA

Personalised recommendations