Asymptotic expansions and fast computation of oscillatory Hilbert transforms

TL;DR

This paper derives asymptotic expansions and develops fast numerical methods for oscillatory Hilbert transforms, especially for large frequencies, with applications to different regimes of the variable x and extensions to Bessel oscillators.

Contribution

It provides the first comprehensive asymptotic analysis and efficient computational techniques for oscillatory Hilbert transforms across various regimes of x.

Findings

Asymptotic expansions in inverse powers of frequency or large requency or different x regimes

Development of efficient numerical algorithms that improve with increasing requency

Extensions to oscillatory transforms involving Bessel functions

Abstract

In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form $$H^{+}(f(t)e^{i\omega t})(x)=-int_{0}^{\infty}e^{i\omega t}\frac{f(t)}{t-x}dt,\qquad \omega>0,\qquad x\geq 0,$$ where the bar indicates the Cauchy principal value and $f$ is a real-valued function with analytic continuation in the first quadrant, except possibly a branch point of algebraic type at the origin. When $x=0$, the integral is interpreted as a Hadamard finite-part integral, provided it is divergent. Asymptotic expansions in inverse powers of $\omega$ are derived for each fixed $x\geq 0$, which clarify the large $\omega$ behavior of this transform. We then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms. Depending on the position of $x$, we classify our discussion into three regimes, namely, $x=\mathcal{O}(1)$ or $x\gg1$, $0<x\ll 1$ and $x=0$. Numerical experiments show that the convergence of the proposed methods greatly improve when the frequency $\omega$ increases. Some extensions to oscillatory Hilbert transforms with Bessel oscillators are briefly discussed as well.