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Zeros and ratio asymptotics for matrix orthogonal polynomials

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Abstract

Ratio asymptotics for matrix orthogonal polynomials with recurrence coefficients A n and B n having limits A and B, respectively, (the matrix Nevai class) were obtained by Durán. In the present paper, we obtain an alternative description of the limiting ratio. We generalize it to recurrence coefficients which are asymptotically periodic with higher periodicity, and/or which are slowly varying as a function of a parameter. Under such assumptions, we also find the limiting zero distribution of the matrix orthogonal polynomials, thus generalizing results by Durán-López-Saff and Dette-Reuther to the non-Hermitian case. Our proofs are based on “normal family” arguments and on the solution of a quadratic eigenvalue problem. As an application of our results, we obtain new explicit formulas for the spectral measures of the matrix Chebyshev polynomials of the first and second kind and derive the asymptotic eigenvalue distribution for a class of random band matrices which generalize the tridiagonal matrices introduced by Dumitriu-Edelman.

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Authors and Affiliations

  1. Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001, Leuven, Belgium

    Steven Delvaux

  2. Department of Mathematics, Ruhr-Universität Bochum, 44780, Bochum, Germany

    Holger Dette

Authors
  1. Steven Delvaux
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  2. Holger Dette
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Correspondence to Steven Delvaux.

Additional information

S. Delvaux is a Postdoctoral Fellow of the Fund for Scientific Research — Flanders (Belgium). His work is supported in part by the Belgian Interuniversity Attraction Pole P06/02.

The work of the authors is supported by the SFB TR12 ”Symmetries and Universality in Mesoscopic Systems”, Teilprojekt C2.

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Delvaux, S., Dette, H. Zeros and ratio asymptotics for matrix orthogonal polynomials. JAMA 118, 657–690 (2012). https://doi.org/10.1007/s11854-012-0047-x

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  • DOI: https://doi.org/10.1007/s11854-012-0047-x

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