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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| Format: | Article |
| Language: | English |
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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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| author | Hongchao Kang Xinping Shao |
| author_facet | Hongchao Kang Xinping Shao |
| author_sort | Hongchao Kang |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-2536781bc92440aa92e8266b145fdf35 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-2536781bc92440aa92e8266b145fdf352025-02-03T05:57:39ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/984834984834Fast Computation of Singular Oscillatory Fourier TransformsHongchao Kang0Xinping Shao1Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, ChinaDepartment of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, ChinaWe 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.http://dx.doi.org/10.1155/2014/984834 |
| spellingShingle | Hongchao Kang Xinping Shao Fast Computation of Singular Oscillatory Fourier Transforms Abstract and Applied Analysis |
| title | Fast Computation of Singular Oscillatory Fourier Transforms |
| title_full | Fast Computation of Singular Oscillatory Fourier Transforms |
| title_fullStr | Fast Computation of Singular Oscillatory Fourier Transforms |
| title_full_unstemmed | Fast Computation of Singular Oscillatory Fourier Transforms |
| title_short | Fast Computation of Singular Oscillatory Fourier Transforms |
| title_sort | fast computation of singular oscillatory fourier transforms |
| url | http://dx.doi.org/10.1155/2014/984834 |
| work_keys_str_mv | AT hongchaokang fastcomputationofsingularoscillatoryfouriertransforms AT xinpingshao fastcomputationofsingularoscillatoryfouriertransforms |