Fast Computation of Singular Oscillatory Fourier Transforms
We consider the problem of the numerical evaluation of singular oscillatory Fourier transforms ∫abx-aαb-xβf(x)eiωxdx, where α>-1 and β>-1. Based on substituting the original interval of integration by the paths of steepest descent, if f is analytic in the complex region G containing [a, b]...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/984834 |
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| Summary: | We consider the problem of the numerical evaluation of singular oscillatory Fourier transforms ∫abx-aαb-xβf(x)eiωxdx, where α>-1 and β>-1. Based on substituting the original interval of integration by the paths of steepest descent, if f is analytic in the complex region G 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. |
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| ISSN: | 1085-3375 1687-0409 |