Asymptotics and numerical approximation of highly oscillatory Hilbert transforms

Abstract

In this paper, we study asymptotics and fast computation of highly oscillatory Hilbert transforms ⨍0+∞f(x)x−τeiωxHν(1)(ωx)dx, where τ > 0 and f is analytic in the complex plane ℜ(z) ≥ 0, which may have an algebraic singularity at the point x=0. By analytic continuation, we first transform Hilbert transforms into the integrals on [0,+∞) with the integrands that don’t oscillate and decay exponentially, and then present the asymptotic behaviour about ω based on asymptotic expansion. For the computation of highly oscillatory Hilbert transforms, we present efficient numerical methods according to the position of τ. To be precise, we construct a Gaussian quadrature rule for it if τ=O(1) or τ ≥ 1. If 0 < τ ≪ 1, we rewrite it as a sum of three integrals, which can be efficiently evaluated by using Chebyshev approximation, Gauss–Laguerre quadrature rule and Meijer G–function. The effectiveness of the proposed methods are demonstrated by several numerical experiments.