Efficient evaluation of oscillatory Bessel Hilbert transforms

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In this paper, we study the asymptotics and evaluation of the oscillatory Bessel Hilbert transform ⨍0∞f(x)x−τJν(ωx)dx with 0<τ<∞. The singularity of the Hilbert transform is transferred to an individual oscillatory integral independent of f(x). For this singular integral, we present two methods. One is the combination of a Filon-type method and a complex integration method, the other is the combination of a Filon-type method and an adaptive Clenshaw–Curtis quadrature. The remaining integral which is nonsingular can be well calculated with a combination of a Filon-type method and a Gauss–Laguerre quadrature. The efficiency and accuracy of the proposed methods are illustrated by numerical examples.

论文关键词:65D32,65D30,Oscillatory Bessel Hilbert transform,Filon-type method,Gauss–Laguerre quadrature,Adaptive Clenshaw–Curtis quadrature

论文评审过程:Received 12 April 2013, Revised 7 July 2013, Available online 13 September 2013.

论文官网地址:https://doi.org/10.1016/j.cam.2013.08.031