Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature

Authors:
A. Deaño;D. Huybrechs;A. B. J. Kuijlaars
Affiliations:
Department of Mathematics, Universidad Carlos III de Madrid, Spain;Department of Computer Science, Katholieke Universiteit Leuven, Belgium;Department of Mathematics, Katholieke Universiteit Leuven, Belgium
Venue:
Journal of Approximation Theory
Year:
2010

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Abstract

In this paper we study the asymptotic behavior of a family of polynomials which are orthogonal with respect to an exponential weight on certain contours of the complex plane. The zeros of these polynomials are the nodes for complex Gaussian quadrature of an oscillatory integral on the real axis with a high order stationary point, and their limit distribution is also analyzed. We show that the zeros accumulate along a contour in the complex plane that has the S-property in an external field. In addition, the strong asymptotics of the orthogonal polynomials are obtained by applying the nonlinear Deift-Zhou steepest descent method to the corresponding Riemann-Hilbert problem.