Error analysis of the extended Filon-type method for highly oscillatory integrals

We investigate the impact of adding inner nodes for a Filon-type method for highly oscillatory quadrature. The error of Filon-type method is composed of asymptotic and interpolation errors, and the interplay between the two varies for different frequencies. We are particularly concerned with two strategies for the choice of inner nodes: Clenshaw–Curtis points and zeros of an appropriate Jacobi polynomial. Once the frequency ω is large, the asymptotic error dominates, but the situation is altogether different when ω ≥ 0 is small. In the first regime, our optimal error bounds indicate that Clenshaw–Curtis points are always marginally better, but this is reversed for small ω; then, Jacobi points enjoy an advantage. The main tool in our analysis is the Peano Kernel theorem. While the main part of the paper addresses integrals without stationary points, we indicate how to extend this work to the case when stationary points are present. Numerical experiments are provided to illustrate theoretical analysis.