Abstract
In this paper, we aim to derive some error bounds for Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals. Thanks to the asymptotics of the coefficients in the Chebyshev series expansions of analytic functions or functions of limited regularities, these bounds are established by the aliasings of Fourier transforms on Chebyshev polynomials together with van der Corput-type lemmas. These errors share the property that the errors decrease with the increase of the frequency ω. Moreover, for fixed ω, the order of the error bound related to the number of interpolation nodes N is attainable, while for fixed N, the order of the error on ω is attainable too, which is verified by some functions of limited regularities. In particular, if the functions are analytic in Bernstein ellipses, then the errors decay exponentially. Furthermore, for large values of ω, the accuracy can be further improved by applying a special Hermite interpolants in the Filon-Clenshaw-Curtis quadrature, which can be efficiently evaluated by the Fast Fourier Transform (FFT) techniques.
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Communicated by: A. Iserles
This paper was supported partly by NSF of China (No.11371376) and partly by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (KRF-2013053358).
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Xiang, S., He, G. & Cho, Y.J. On error bounds of Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals. Adv Comput Math 41, 573–597 (2015). https://doi.org/10.1007/s10444-014-9377-9
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DOI: https://doi.org/10.1007/s10444-014-9377-9