On the numerical quadrature of highly-oscillating integrals II: Irregular oscillators

In this paper we set out to understand Filon-type quadrature of highly-oscillating integrals of the form ∫ 1 0 f (x)e iωg(x)dx , where g is a real-valued function and ω 1. Employing ad hoc analysis, as well as perturbation theory, we demonstrate that for most functions g of interest the moments behave asymptotically according to a specific model that allows for an optimal choice of quadrature nodes. Filontype methods that employ such quadrature nodes exhibit significantly faster decay of the error for high frequencies ω. Perhaps counterintuitively, as long as optimal quadrature nodes are used, rapid oscillation leads to significantly more precise and more affordable quadrature.