×

Strong asymptotics of orthogonal polynomials with respect to exponential weights. (English) Zbl 1026.42024

In this excellent paper the authors study asymptotics for orthogonal polynomials with respect to the weights \[ w(x)dx=e^{-Q(x)}dx \] on the real line, with \(Q(x)\) an even polynomial of degree \(2m\) with positive leading coefficients
The main results cover
a. asymptotics for the leading and recurrence coefficients (Theorem 2.1),
b. Plancherel-Rotach asymptotics on the whole complex plane (Theorem 2.2),
c. asymptotic location of the zeros (Theorem 2.3).
The deep results are derived through recently developed methods and a reformulation as a Riemann-Hilbert problem due to A. S. Fokas, A. R. Its and A. V. Kitaev [Commun. Math. Phys. 142, 313-344 (1991; Zbl 0742.35047); ibid. 147, 395-430 (1992; Zbl 0760.35051)].
The solution of this Riemann-Hilbert problem is then subjected to a series of transformations, leading to deep asymptotic results.
The technical and delicate operations are described in detail and give the reader a good insight in the different techniques needed. In view of the intricacies of the methods, the length of the paper is just about right.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
30E25 Boundary value problems in the complex plane
35Q15 Riemann-Hilbert problems in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] ; Handbook of mathematical functions, with formulas, graphs, and mathematical tables. Dover, New York, 1966.
[2] Orthogonal polynomials associated with exponential weights. Thesis, Ohio State University, 1985.
[3] Bauldry, J Approx Theory 63 pp 225– (1990)
[4] Bauldry, Pacific J Math 133 pp 209– (1988)
[5] Beals, Comm Pure Appl Math 37 pp 39– (1984)
[6] ; ; Direct and inverse scattering on the line. Mathematical Surveys and Monographs, 28. American Mathematical Society, Providence, R.I., 1988.
[7] ; Asymptotics of orthogonal polynomials and universality in matrix models. Preprint, 1996.
[8] Bonan, J Approx Theory 63 pp 210– (1990)
[9] ; Ladder operators and differential equations for orthogonal polynomials. Preprint, 1997.
[10] ; Factorization of matrix functions and singular integral operators. Operator Theory: Advances and Applications, 3. Birkhäuser, Basel-Boston, 1981.
[11] Criscuolo, J Math Anal Appl 189 pp 256– (1995)
[12] Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, 3. Courant Institute, New York, 1999.
[13] Deift, Comm Pure Appl Math 49 pp 35– (1996)
[14] Deift, J Approx Theory 95 pp 388– (1998)
[15] Deift, Internat Math Res Notices pp 759– (1997)
[16] Deift, Comm Pure Appl Math 52 (1999)
[17] Deift, Internat Math Res Notices pp 286– (1997)
[18] Deift, Ann of Math (2) 137 pp 295– (1993)
[19] Deift, Comm Pure Appl Math 48 pp 277– (1995)
[20] Ercolani, Comm Math Phys 183 pp 119– (1997)
[21] ; ; The KdV zero dispersion limit via Dirichlet spectra and Weyl functions. Preprint, 1997.
[22] Fokas, Comm Math Phys 142 pp 313– (1991)
[23] Fokas, Comm Math Phys 147 pp 395– (1992)
[24] ; One-dimensional linear singular integral equations. I. Introduction. Translated from the 1979 German translation by Bernd Luderer and Steffen Roch and revised by the authors. Operator Theory: Advances and Applications, 53. Birkhäuser, Basel, 1992.
[25] ; One-dimensional linear singular integral equations. Vol. II. General theory and applications. Translated from the 1979 German translation by S. Roch and revised by the authors. Operator Theory: Advances and Applications, 54. Birkhäuser, Basel, 1992.
[26] Lax, Comm Pure Appl Math 36 pp 253– (1983)
[27] Lax, Comm Pure Appl Math 36 pp 571– (1983)
[28] Lax, Comm Pure Appl Math 36 pp 809– (1983)
[29] Levin, Constr Approx 8 pp 463– (1992)
[30] Strong asymptotics for extremal errors and polynomials associated with Erdös-type weights. Pitman Research Notes in Mathematics Series, 202. Longman, Harlow; copublished in the United States with John Wiley, New York, 1989.
[31] Lubinsky, Acta Appl Math 33 pp 121– (1993)
[32] Lubinsky, Constr Approx 4 pp 65– (1988)
[33] ; Strong asymptotics for extremal polynomials associated with weights on R. Lecture Notes in Mathematics, 1305. Springer, Berlin-New York, 1988. · Zbl 0647.41001
[34] Magnus, J Approx Theory 46 pp 65– (1986)
[35] Máté, J London Math Soc (2) 33 pp 303– (1986)
[36] Mhaskar, J Approx Theory 63 pp 238– (1990)
[37] Mhaskar, Trans Amer Math Soc 285 pp 203– (1984)
[38] Nevai, SIAM J Math Anal 15 pp 1177– (1984)
[39] Plancherel, Comment Math Helv 1 pp 227– (1929)
[40] Rakhmanov, Mat Sb (NS) 119 pp 163– (1982)
[41] Rakhmanov, Math Sb (NS) 47 pp 155– (1984)
[42] Strong asymptotics for orthogonal polynomials. Methods of approximation theory in complex analysis and mathematical physics (Leningrad, 1991), 71-97. Lecture Notes in Math, 1550, Springer, Berlin, 1993.
[43] One-dimensional perturbations of a differential operator, and the inverse scattering problem. (Russian) Problems in mechanics and mathematical physics (Russian), 279-296, 298. Izdat. ?Nauka,? Moscow, 1976.
[44] ; Logarithmic potentials with external fields. Appendix B by Thomas Bloom. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 316. Springer, Berlin, 1997. · Zbl 0881.31001
[45] Sheen, J Approx Theory 50 pp 232– (1987)
[46] Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.
[47] Orthogonal polynomials. American Mathematical Society Colloquium Publications, Vol. 23. Revised ed. American Mathematical Society, Providence, R.I., 1959.
[48] Weighted approximation with varying weight. Lecture Notes in Mathematics, 1569. Springer, Berlin, 1994.
[49] Van Assche, Rocky Mountain J Math 19 pp 39– (1989)
[50] Zhou, SIAM J Math Anal 20 pp 966– (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.