• Eli Levin
  • Doron S. Lubinsky
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


We consider varying weights \( \left\{e^{-2nQ_{n}}\right\}\) and establish bounds and asymptotics for their orthonormal polynomials, and associated quantities, when \( \left\{ Q_{n}\right\}\) are convex and \( \left\{ Q_{n}^{\prime }\right\} \) satisfy a uniform Hölder condition of appropriate order. We deduce universality limits for related unitary ensembles, and asymptotics for entropy integrals.


  1. 3.
    A. I. Aptekarev, R. Khabibullin, Asymptotic Expansions for Polynomials Orthogonal with respect to a Complex Non-Constant Weight Function, Trans. Moscow Math. Soc., 68(2007), 1–37.CrossRefzbMATHGoogle Scholar
  2. 7.
    D. Dai, M. E. H. Ismail, X.-S. Wang, Plancherel-Rotach Asymptotic Expansion for Some Polynomials from Indeterminate Moment Problems, Constr Approx, 40(2014), 61–104.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 9.
    P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Institute Lecture Notes, Vol. 3, New York University Press, New York, 1999.Google Scholar
  4. 11.
    P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Venakides, X. Zhou, Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (1999), 1491–1552.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 12.
    P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Venakides, X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), 1335–1425.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 14.
    J. S. Geronimo, D. Smith, and W. Van Assche, Strong asymptotics for orthogonal polynomials with regularly and slowly varying recurrence coefficients, J. Approx. Theory, 72(1993), 141–158.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 15.
    J. S. Geronimo and W. Van Assche, Relative asymptotics for orthogonal polynomials with unbounded recurrence coefficients, J. Approx. Theory, 62, (1990), 47–69.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 18.
    A. V. Komlov, S. P. Suetin, An Asymptotic Formula for Polynomials Orthonormal with respect to a Varying Weight, Trans. Moscow Math. Soc., 73(2012), 139–159.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 23.
    A. B. Kuijlaars and M. Vanlessen, Universality for Eigenvalue Correlations from the Modified Jacobi Unitary Ensemble, International Mathematics Research Notices, 30(2002), 1575–1600.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 25.
    Eli Levin and D. S. Lubinsky, Orthogonal Polynomials for Exponential Weights,Springer, New York, 2001.Google Scholar
  11. 27.
    Eli Levin and D. S. Lubinsky, Universality Limits in the Bulk for Varying Measures, Advances in Mathematics, 219(2008), 743–779.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 32.
    D. S. Lubinsky, E. B. Saff, Strong Asymptotics for Extremal Polynomials Associated with Weights on \(\mathbb {R}\), Springer Lecture Notes in Mathematics, Vol. 1305, Springer, Berlin, 1988.Google Scholar
  13. 34.
    K. T.-R McLaughlin and P. D. Miller, The \(\bar {\partial }\) Steepest Descent Method and the Asymptotic Behavior of Polynomials Orthogonal on the Unit Circle with Fixed and Exponentially Varying Nonanalytic Weights, Int. Math. Res. Notices, (2006), Article ID 48673, 78 pages.Google Scholar
  14. 35.
    K. T.-R McLaughlin and P. D. Miller, The \(\bar {\partial }\) Steepest Descent Method for Orthogonal Polynomials on the Real Line with Varying Weights, Int. Math. Res. Notices, (2008), Article ID rnn075, 66 pages.Google Scholar
  15. 36.
    H. N. Mhaskar and E. B. Saff, Extremal Problems for Polynomials with Exponential Weights, Trans. Amer. Math. Soc., 285(1984), 203–234.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 38.
    H. N. Mhaskar and E. B. Saff, Where does the L p Norm of a Weighted Polynomial Live?, Trans. Amer. Math. Soc., 303(1987), 109–124.Google Scholar
  17. 40.
    P. Nevai, Geza Freud, Orthogonal Polynomials and Christoffel Functions: A Case Study, J. Approx. Theory, 48(1986), 3–167.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 41.
    M. Plancherel and W. Rotach, Sur les valeurs asymptotiques des polynomes d’Hermite \(H_{n}\left ( x\right ) =\left ( -1\right ) ^{n}e^{x^{2}/2}\frac {d^{n}}{dx^{n}}\left ( e^{-x^{2}/2}\right ) \), Commentarii Mathematici Helvetici, 1(1929), 227–254.Google Scholar
  19. 42.
    E. A. Rakhmanov, On Asymptotic Properties of Polynomials Orthogonal on the Real Axis, Math. USSR. Sbornik, 47(1984), 155–193.CrossRefzbMATHGoogle Scholar
  20. 43.
    E. A. Rakhmanov, Strong Asymptotics for Orthogonal Polynomials Associated with Exponential Weights on \(\mathbb {R}\), (in) Methods of Approximation Theory in Complex Analysis and Mathematical Physics, (eds. A. A. Gonchar and E. B. Saff), Nauka, Moscow, 1992, pp. 71–97.Google Scholar
  21. 44.
    E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer, New York, 1997.CrossRefzbMATHGoogle Scholar
  22. 47.
    G. Szegő, A Hankel-féle formákról, Mathematikai és Természettudomanyi Értesito, 36(1918), 497–538.Google Scholar
  23. 48.
    G. Szegő, Über Orthogonalsysteme von Polynomen, Mathematische Zeitschrift, 4(1919), 139–151.Google Scholar
  24. 51.
    V. Totik, Weighted Approximation with Varying Weight, Springer Lecture Notes in Mathematics, Vol. 1569, Springer, Berlin, 1994.CrossRefGoogle Scholar
  25. 52.
    V. Totik, Asymptotics for Christoffel Functions with Varying Weights, Advances in Applied Mathematics, 25(2000), 322–351.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 55.
    W. Van Assche, Asymptotics for Orthogonal Polynomials, Lecture Notes in Mathematics, Vol. 1265, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar
  27. 56.
    M. Vanlessen, Strong Asymptotics of Laguerre-type Orthogonal Polynomials and Applications in Random Matrix Theory, Constr. Approx., 25 (2007), 125–175.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 57.
    Z. Wang and R. Wong, Asymptotic expansions for second-order linear difference equations with a turning point, Numer. Math., 94, (2003), 147–194.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Eli Levin
    • 1
  • Doron S. Lubinsky
    • 2
  1. 1.Department of MathematicsOpen University of IsraelTel-AvivIsrael
  2. 2.MathematicsGeorgia Institution of TechnologyAtlantaUSA

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