Fast Computation of Singular Oscillatory Fourier Transforms. (17th July 2014)

• Record Type: Journal Article
• Authors: Kang, Hongchao; Shao, Xinping
• Other Names: Bai Chuanzhi Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2014(2014)
• Publication Date: 2014-07-17
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2014/984834
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : We consider the problem of the numerical evaluation of singular oscillatory Fourier transforms ∫ a b ‍ x - a α b - x β f ( x ) e i ω x d x, whereα > - 1 and β > - 1 . Based on substituting the original interval of integration by the paths of steepest descent, iff is analytic in the complex regionG containing [a, b ], 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.