Citation:
Journal of Approximation Theory, October 2014, Vol.186, October 2014, pp. 33–63
ISSN:
0021-9045
DOI:
10.1016/j.jat.2014.07.004
Sponsor:
The author gratefully acknowledges financial support from projects FWO G.0617.10 and
FWO G.0641.11, funded by FWO (Fonds Wetenschappelijk Onderzoek, Research Fund Flanders, Belgium), and projects MTM2012-34787 and MTM2012-36732-C03-01, from the Spanish Ministry of Economy and Competitivity. The author also thanks Daan Huybrechs and Pablo Rom´an for many stimulating and useful discussions on the topic and scope of this paper, and the two anonymous referees for comments, corrections and suggestions to improve its presentation.
Project:
Gobierno de España. MTM2012-34787 Gobierno de España. MTM2012-36732-C03-01
Keywords:
Orthogonal polynomials in the complex plane
,
Strong asymptotics
,
Zero distribution
,
Logarithmic potential theory
,
S-property
,
Steepest descent method
,
Riemann-Hilbert problem
We consider polynomials pω
n (x) that are orthogonal with respect to the oscillatory weight w(x) = eiωx
on [−1, 1], where ω > 0 is a real parameter. A first analysis of pω
n (x) for large values of ω was carried out in
Asheim et al. (2014), in connection wWe consider polynomials pω
n (x) that are orthogonal with respect to the oscillatory weight w(x) = eiωx
on [−1, 1], where ω > 0 is a real parameter. A first analysis of pω
n (x) for large values of ω was carried out in
Asheim et al. (2014), in connection with complex Gaussian quadrature rules with uniform good properties
in ω. In this contribution we study the existence, asymptotic behavior and asymptotic distribution of the
roots of pω
n (x) in the complex plane as n → ∞. The parameter ω grows with n linearly. The tools used
are logarithmic potential theory and the S-property, together with the Riemann–Hilbert formulation and the
Deift–Zhou steepest descent method.[+][-]