Publication: Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval
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2014-10
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Elsevier
Abstract
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 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.
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Keywords
Orthogonal polynomials in the complex plane, Strong asymptotics, Zero distribution, Logarithmic potential theory, S-property, Steepest descent method, Riemann-Hilbert problem
Bibliographic citation
Journal of Approximation Theory, October 2014, Vol.186, October 2014, pp. 33–63