Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature. (arXiv:1001.2219v1 [math.CA])

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 is obtained by applying the nonlinear Deift--Zhou
steepest descent method to the corresponding Riemann--Hilbert problem.

Published by

Original source : http://arxiv.org/abs/1001.2219...