On complex orthogonal polynomials related with Gaussian quadrature of oscillatory integrals

Analysis Seminar
Wednesday, April 28, 2010 - 14:00
1 hour (actually 50 minutes)
Skiles 269
Universidad Carlos III de Madrid (Spain)
We present results on the asymptotic behavior of a family of polynomials which are orthogonal with respect to an exponential weight on certain contours of the complex plane. Our motivation comes from the fact that the zeros of these polynomials are the nodes for complex Gaussian quadrature of an oscillatory integral defined on the real axis and having  a high order stationary point. The limit distribution of these zeros is also analyzed, and we show that they accumulate along a contour in the complex plane that has the S-property in the presence of an external field. Additionally, the strong asymptotics of the orthogonal polynomials is obtained by applying the nonlinear Deift--Zhou steepest descent method to the corresponding Riemann--Hilbert problem. This is joint work with D. Huybrechs and A. Kuijlaars, Katholieke Universiteit Leuven (Belgium).