Asymptotic expansions and fast computation of oscillatory Hilbert transforms

作者：Wang Haiyong^*; Zhang Lun; Huybrechs Daan

来源：Numerische Mathematik, 2013, 123(4): 709-743.

DOI：10.1007/s00211-012-0501-9

In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form <Equation ID="Equa1"> <MediaObject> </MediaObject> </Equation>where the bar indicates the Cauchy principal value and is a real-valued function with analytic continuation in the first quadrant, except possibly a branch point of algebraic type at the origin. When , the integral is interpreted as a Hadamard finite-part integral, provided it is divergent. Asymptotic expansions in inverse powers of are derived for each fixed , which clarify the large behavior of this transform. We then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms. Depending on the position of , we classify our discussion into three regimes, namely, or , and . Numerical experiments show that the convergence of the proposed methods greatly improve when the frequency increases. Some extensions to oscillatory Hilbert transforms with Bessel oscillators are briefly discussed as well.

• 出版日期2013-4