zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

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