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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 984834, 8 pages
http://dx.doi.org/10.1155/2014/984834
Research Article

Fast Computation of Singular Oscillatory Fourier Transforms

Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China

Received 1 April 2014; Accepted 3 July 2014; Published 17 July 2014

Academic Editor: Chuanzhi Bai

Copyright © 2014 Hongchao Kang and Xinping Shao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the problem of the numerical evaluation of singular oscillatory Fourier transforms , where . Based on substituting the original interval of integration by the paths of steepest descent, if is analytic in the complex region containing [ , ], the computation of integrals can be transformed into the problems of integrating two integrals on [0, ∞) with the integrand that does not oscillate and decays exponentially fast, which can be efficiently computed by using the generalized Gauss Laguerre quadrature rule. The efficiency and the validity of the method are demonstrated by both numerical experiments and theoretical results. More importantly, the presented method in this paper is also a great improvement of a Filon-type method and a Clenshaw-Curtis-Filon-type method shown in Kang and Xiang (2011) and the Chebyshev expansions method proposed in Kang et al. (2013), for computing the above integrals.