Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature
This provides a theoretical foundation for complex Gaussian quadrature, relevant to numerical analysis of oscillatory integrals.
The paper analyzes the asymptotic zero distribution of complex orthogonal polynomials used in Gaussian quadrature for oscillatory integrals, showing that zeros accumulate along a contour with the S-property and deriving strong asymptotics via the Deift-Zhou method.
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.