## 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

### Citations:

Zbl 0760.35051; Zbl 0742.35047
Full Text:

### 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)
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