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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:
42C05General theory of orthogonal functions and polynomials
30E25Boundary value problems, complex analysis
35Q15Riemann-Hilbert problems