Mathematics of Computation

Asymptotics of recurrence coefficients for orthonormal polynomials on the line--Magnus's method revisited

Author(s): S. B. Damelin.
Journal: Math. Comp. 73 (2004), 191-209.
MSC (2000): Primary 45M05, 33D45, 41A10, 65Q05, 42B05, 30D20, 35Q15, 15A42, 15A60
Posted: July 28, 2003
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Abstract: We use Freud equations to obtain the main term in the asymptotic expansion of the recurrence coefficients associated with orthonormal polynomials

for weights $w=W\exp(-Q)$ on the real line where

is an even polynomial of fixed degree with nonnegative coefficients or where $Q(x) =\exp(x^{2m}),\, m\geq 1$ . Here $W(x)=\vert x\vert^{\rho}$ for some real $\rho>-1$ .

1.: W. Bauldry, A. Máté and P. Nevai, Asymptotic expansions of recurrence coefficients of asymmetric Freud polynomials, in ``Approximation Theory V'' (C. Chui, L. Schumaker and J. Ward, eds.), Academic Press, New York, 1987, pp. 251-254.
2.: -, Asymptotics for the solutions of systems of smooth recurrence equations, Pacific J. Math. 133 (1988), pp. 209-227. MR 89m:39002
3.: P. Bleher and A. Its, Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem and universality in the matrix model, Annals of Mathematics, 150 (1999), 185-266. MR 2000k:42003
4.: P. Deift, T. Kriecherbauer, K. McLaughlin, S. Venakides and X. Zhou, Strong asymptotics of orthogonal polynomials with respect to exponential weights, Commun. Pure Appl. Math. 52 (1999), 1491-1552. MR 2001f:42037
5.: W. Fleming, Functions of Several Variables, Texts in Mathematics, Springer-Verlag, 1987. MR 54:10514
6.: G. Freud, On the greatest zero of orthogonal polynomials, J. Approx. Theory 46 (1986), pp. 15-23. MR 87f:42057
7.: -, On the coefficients in the recursion formulae of orthogonal polynomials, Proc. Roy. Irish Acad, Sect. A (1) 76 (1976), pp. 1-6. MR 54:7913
8.: -, Orthogonal Polynomials, Akadémiai Kiadó, Budapest, 1971.
9.: A. L. Levin and D. S. Lubinsky, Orthogonal Polynomials for Exponential Weights, Springer-Verlag, 2001. MR 2002k:41001
10.: D. S. Lubinsky, H. N. Mhaskar, and E. B. Saff, A proof of Freud's conjecture for exponential weights, Constr. Approx. 4 (1988), pp. 65-83. MR 89a:42034
11.: A. P. Magnus, On Freud's equations for exponential weights, J. Approx. Theory 46 (1986), pp. 65-99. MR 87h:42039
12.: -, A proof of Freud's about orthogonal polynomials related to $\vert x\vert^{\rho}\exp(-x^{2m})$ . ``Orthogonal Polynomials and their Applications'' (C. Brezinski et al., eds.), Lecture Notes in Mathematics, Vol. 1171, Springer-Verlag, Berlin, 1985. MR 87g:42040
13.: -, Freud's equations for orthogonal polynomials as discrete Painlevé equations, London Math. Soc. Lecture Note Ser., no. 255, pp. 228-243. MR 2000k:42036
14.: H. N. Mhaskar, Introduction to the Theory of Weighted Approximation, World Scientific, 1996. MR 98i:41014
15.: P. Nevai, Personal communication.
16.: E. A. Rakhmanov, Strong asymptotics for orthogonal polynomials, ``Methods of Approximation Theory in Complex Analysis and Mathematical Physics'' (A. A. Gonchar and E. B. Saff, eds.), Lecture Notes in Mathematics 1550, Springer-Verlag, Berlin, 1993, pp. 71-97. MR 96a:42031
17.: V. Totik, Weighted approximation with varying weights, Lecture Notes in Mathematics 1300, Springer-Verlag, Berlin, 1994. MR 96f:41002
18.: W. Van Assche, Weak convergence of orthogonal polynomials, Indag. Math. N.S. 6 (1995), pp. 7-23. MR 96a:42033

S. B. Damelin
Affiliation: Department of Mathematics and Computer Science, Georgia Southern University, P. O. Box 8093, Statesboro, Georgia 30460
Email: damelin@gsu.cs.gasou.edu

DOI: 10.1090/S0025-5718-03-01553-9
PII: S 0025-5718(03)01553-9
Keywords: Asymptotics, entire functions of finite and infinite order, Erd\H{o}s weights, Freud weights, orthogonal polynomials, recurrence coefficients
Received by editor(s): September 7, 2001
Received by editor(s) in revised form: June 19, 2002
Posted: July 28, 2003
Copyright of article: Copyright 2003, American Mathematical Society

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