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1 References 1. G. Akemann, J. Baik, P. Di Francesco (editors), The Oxford Handbook of Random Matrix Theory, Oxford University Press, Oxford, G. Anderson, A. Guionnet, O. Zeitouni, An Introduction to Random Matrices, Cambridge Studies in Advanced Mathematics, 118. Cambridge University Press, Cambridge, 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), A. I. Aptekarev, J. S. Dehesa, A. Martinez-Finkelshtein, Asymptotics of Orthogonal Polynomial s Entropy, J. Comp. Appl. Math., 233(2010), J. Baik, L. Li, T. Kriecherbauer, K. McLaughlin, C. Tomei, Proceedings of the Conference on Integrable Systems, Random Matrices and Applications, Contemporary Mathematics, Vol. 458, Amer. Math. Soc., Providence, B. Beckermann, A. Martinez-Finkelshtein, E. A. Rakhmanov, F. Wielonsky, Asymptotic Upper Bounds for the Entropy of Orthogonal Polynomials in the SzegőClass,J.Math.Phys., 45(2004), D. Dai, M. E. H. Ismail, X.-S. Wang, Plancherel-Rotach Asymptotic Expansion for Some Polynomials from Indeterminate Moment Problems, Constr Approx, 40(2014), J. S. Dehesa, V. S. Buyarov, A. Martinez-Finkelshtein, E. B. Saff, Asymptotics of the Information Entropy for Jacobi and Laguerre Polynomials with Varying Weights, J. Approx. Theory, 99(1999), P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Institute Lecture Notes, Vol. 3, New York University Press, New York, P. Deift, D. Gioev, Random Matrix Theory: Invariant Ensembles and Universality, Courant Institute Lecture Notes, Vol. 18, New York University Press, New York, 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), 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), P. J. Forrester, Log-gases and Random matrices, Princeton University Press, Princeton, The Author(s) 2018 E. Levin, D.S. Lubinsky, Bounds and Asymptotics for Orthogonal Polynomials for Varying Weights, SpringerBriefs in Mathematics, 165

2 166 References 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), J. S. Geronimo and W. Van Assche, Relative asymptotics for orthogonal polynomials with unbounded recurrence coefficients, J. Approx. Theory, 62, (1990), I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, San Diego, J. Gustavsson, Gaussian Fluctuations of Eigenvalues in the GUE, Annales Institut H. Poincare, 41(2005), A. V. Komlov, S. P. Suetin, An Asymptotic Formula for Polynomials Orthonormal with respect to a Varying Weight, Trans. Moscow Math. Soc., 73(2012), A. V. Komlov, S. P. Suetin, An Asymptotic Formula for Polynomials Orthonormal with respect to a Varying Weight II, Sbornik Mathematics, 205(2014), A. B. Kuijlaars, Riemann-Hilbert Analysis for Orthogonal Polynomials, (in) Orthogonal Polynomials and Special Functions, Leuven 2002" (E. Koelink and W. Van Assche, eds.), Lect. Notes Math. 1817, Springer-Verlag, pp A. B. Kuijlaars, K. T-R McLaughlin, Riemann-Hilbert Analysis for Laguerre Polynomials with Large Negative Parameter, Comput. Meth. Funct. Theory, 1(2001), A. Kuijlaars, K. T-R McLaughlin, W. Van Assche, and M. Vanlessen, The Riemann- Hilbert approach to strong asymptotics for orthogonal polynomials on Œ 1; 1, Advances in Mathematics, 188 (2004), A. B. Kuijlaars and M. Vanlessen, Universality for Eigenvalue Correlations from the Modified Jacobi Unitary Ensemble, International Mathematics Research Notices, 30(2002), Eli Levin and D. S. Lubinsky, Christoffel Functions, Orthogonal Polynomials, and Nevai s Conjecture for Freud Weights, Constr. Approx., 8(1992), Eli Levin and D. S. Lubinsky, Orthogonal Polynomials for Exponential Weights, Springer, New York, Eli Levin and D. S. Lubinsky, Asymptotics for Entropy Integrals Associated with Exponential Weights, J. Comp. Appl. Math., 156(2003), Eli Levin and D. S. Lubinsky, Universality Limits in the Bulk for Varying Measures, Advances in Mathematics, 219(2008), D. S. Lubinsky, A New Approach to Universality Limits involving Orthogonal Polynomials, Annals of Mathematics, 170(2009), D. S. Lubinsky, Bulk Universality Holds in Measure for Compactly Supported Measures, J. d Analyse de Mathematique, 116(2012), D. S. Lubinsky, Gaussian Fluctuations of Eigenvalues of Random Hermitian Matrices Associated with Fixed and Varying Weights, Random Matrices: Theory and Applications, 5(2016), No. 3, , 31 pages, /S X. 31. D. S. Lubinsky, An Update on Local Universality Limits for Correlation Functions generated by Unitary Ensembles, SIGMA, 12(2016), 078, 36 pages. 32. D. S. Lubinsky, E. B. Saff, Strong Asymptotics for Extremal Polynomials Associated with Weights on R, Springer Lecture Notes in Mathematics, Vol. 1305, Springer, Berlin, G. Mastroianni, G. V. Milovanovic, Interpolation Processes, Basic Theory and Applications, Springer, New Berlin, K. T.-R McLaughlin and P. D. Miller, The N@ 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. 35. K. T.-R McLaughlin and P. D. Miller, The N@ Steepest Descent Method for Orthogonal Polynomials on the Real Line with Varying Weights, Int. Math. Res. Notices, (2008), Article ID rnn075, 66 pages. 36. H. N. Mhaskar and E. B. Saff, Extremal Problems for Polynomials with Exponential Weights, Trans. Amer. Math. Soc., 285(1984),

3 References H. N. Mhaskar and E. B. Saff, Where does the Sup Norm of a Weighted Polynomial Live?, Constr. Approx., 1(1985), H. N. Mhaskar and E. B. Saff, Where does the L p Norm of a Weighted Polynomial Live?, Trans. Amer. Math. Soc., 303(1987), P. Nevai, Orthogonal Polynomials, Memoirs of the American Math. Soc., 18(1979), No P. Nevai, Geza Freud, Orthogonal Polynomials and Christoffel Functions: A Case Study, J. Approx. Theory, 48(1986), M. Plancherel and W. Rotach, Sur les valeurs asymptotiques des polynomes d Hermite H n.x/ D. 1/ n e x2 =2 d n e dx n x2 =2, Commentarii Mathematici Helvetici, 1(1929), E. A. Rakhmanov, On Asymptotic Properties of Polynomials Orthogonal on the Real Axis, Math. USSR. Sbornik, 47(1984), E. A. Rakhmanov, Strong Asymptotics for Orthogonal Polynomials Associated with Exponential Weights on R, (in) Methods of Approximation Theory in Complex Analysis and Mathematical Physics, (eds. A. A. Gonchar and E. B. Saff), Nauka, Moscow, 1992, pp E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer, New York, B. Simon, Two Extensions of Lubinsky s Universality Theorem, J. d Analyse Mathematique, 105 (2008), B. Simon, Szegő s Theorem and its descendants, Princeton University Press, Princeton, G. Szegő, A Hankel-féle formákról, Mathematikai és Természettudomanyi Értesito, 36(1918), G. Szegő, Über Orthogonalsysteme von Polynomen, Mathematische Zeitschrift, 4(1919), G. Szegő, Orthogonal Polynomials, American Mathematical Society, Providence, E. C. Titchmarsh, Fourier Integrals, Chelsea Publishing, New York, V. Totik, Weighted Approximation with Varying Weight, Springer Lecture Notes in Mathematics, Vol. 1569, Springer, Berlin, V. Totik, Asymptotics for Christoffel Functions with Varying Weights, Advances in Applied Mathematics, 25(2000), V. Totik, Universality and fine zero spacing on general sets, Arkiv för Matematik, 47(2009), V. Totik, Universality Under Szegő s Condition, Canadian Math. Bulletin, 59 (2016), W. Van Assche, Asymptotics for Orthogonal Polynomials, Lecture Notes in Mathematics,Vol. 1265, Springer-Verlag, Berlin, M. Vanlessen, Strong Asymptotics of Laguerre-type Orthogonal Polynomials and Applications in Random Matrix Theory, Constr. Approx., 25 (2007), Z. Wang and R. Wong, Asymptotic expansions for second-order linear difference equations with a turning point, Numer. Math., 94, (2003), Deng Zhang, Gaussian Fluctuations of Eigenvalues in Log-gas Ensemble: Bulk Case I, Acta Math. Sinica, 31(2015), Deng Zhang, Gaussian Fluctuations of Eigenvalues in the Unitary Ensemble, manuscript. 60. A. Zygmund, Trigonometric Series, Vols. 1,2, Second Paperback edition, Cambridge University Press, Cambridge, 1988.

4 Index Symbols A n;m,12 A n.x/, 53 B n;m,12 D.F n I z/,11 F n./, 11 F Qn, F Qn ;r,14 I n,6 K n. n ; x; y/, K n.x; y/, 3 L n;r,14 Q n.x/, 6 S n.x/, 93 U n ; U n;r,14 V Qn;r,14.f n I /, 118.u/, 69, 84 n;j ;.1/ n;j, 69, 84 NQ n.x; t/, 53 ˇn;r,14 ı n;r,14 `jn, 143 n;m,2 n.; x/; n;p.; x/, 3,37 tj,68 n.u/, 0 n P T,29 R.n/ m, 153! Q,4.z/,11 n;m, 128 n.t/,93 Qn ; Qn ;r,14, 8 QK n.x; y/, 154 a n;r,13 c n,6 d n,6 f n./, 117 g.z/; g.z; a/,24 p n;k,2 w n;b,93 x jn,10 y jn, 125 B Bernstein-Szego weight, 93, 117 approximation by, 93 C Christoffel functions, 3 asymptotics, 11, 124 bounds, 37 D discretization of potentials, 67 The Author(s) 2018 E. Levin, D.S. Lubinsky, Bounds and Asymptotics for Orthogonal Polynomials for Varying Weights, SpringerBriefs in Mathematics, 169

5 170 Index E entropy integrals, 155, 156 equilibrium density, 4, 13 bounds, 15 equilibrium measure, 4 equilibrium potential, 4, 14 external field, 3 O orthogonal polynomials asymptotics in the plane, 124 bounds, 9, 53 derivatives, 10 leading coefficients, 123 pointwise asymptotics, 10, 123 F fundamental polynomials of Lagrange interpolation, 143 G Green s function, 30 H Hilbert transform, 16 K Korous method, 129 L local fluctuation of eigenvalues, 155 Lp Christoffel function, 37 M Markov-Bernstein inequalities, 63 Mhaskar-Rakhmanov-Saff numbers, 6, 13 Mhaskar-Saff identity, 29 R random matrices, 153 recurrence coefficients asymptotics, 12 reproducing kernel, 3 restricted range inequality, 29 S sup norm Christoffel function, 39 Szego function, 11, 117 argument of, 118 U universality limit, 154, 155 Z zeros of orthogonal polynomials, 47 spacing, 10, 11, 125, 144

Scaling Limits of Polynomials and Entire Functions of Exponential Type

Scaling Limits of Polynomials and Entire Functions of Exponential Type D. S. Lubinsky Abstract The connection between polynomials and entire functions of exponential type is an old one, in some ways harking