Complex generalized Gauss–Radau quadrature rules for Hankel transforms of integer order

Abstract

Complex Gaussian quadrature rules for oscillatory integral transforms have the advantage that they can achieve optimal asymptotic order. However, their existence for Hankel transform can only be guaranteed when the order of the transform belongs to $[0,1/2]$. In this paper we introduce a new family of Gaussian quadrature rules for Hankel transforms of integer order. We show that, if adding certain value and derivative information at the left endpoint, then complex generalized Gauss–Radau quadrature rules that guarantee existence can be constructed and their nodes and weights can be calculated from a half-size Gaussian quadrature rule with respect to the generalized Prudnikov weight function. Orthogonal polynomials that are closely related to such quadrature rules are investigated and their existence for even degrees is proved. Numerical experiments are presented to show the performance of the proposed rules.