Asymptotic expansions and fast computation of oscillatory Hilbert transforms

In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form $$\begin{aligned} H^{+}(f(t)e^{i\omega t})(x)=-\!\!\!\!\!\!\int \nolimits _{\!\!\!0}^{\infty }e^{i\omega t}\frac{f(t)}{t-x}\,dt,\quad \omega >0,\quad x\ge 0, \end{aligned}$$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 $$x\ge 0$$, 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 $$x\gg 1$$, $$0<x\ll 1$$ 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.