Reproducing kernel function-based formulation for highly oscillatory integrals

Abstract

Reproducing-kernel functions are effective approximating tools for interpolation of various types of functions regardless of the troublesome sensitivity to shape parameters like that of Radial Basis Functions (RBFs). In the current work, a stable algorithm based on reproducing-kernel functions is proposed for numerical evaluation of oscillatory integrals with or without stationary phase. Reproducing-kernel functions, defined on a real Hilbert space, serve as basis functions in the Levin formulation. The proposed algorithm provides accurate approximation on both uniformly distributed and scattered data points in similar pattern to that of RBFs. High-resolution integration techniques based on wavelets are combined with reproducing kernel functions to evaluate oscillatory integrals with stationary phase. Theoretical error bounds of the new algorithm are derived. Several test cases are included to demonstrate accuracy and efficiency of the proposed algorithm.