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Mathematics of Computation

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Asymptotic expansion and quadrature of composite highly oscillatory integrals


Authors: Arieh Iserles and David Levin
Journal: Math. Comp. 80 (2011), 279-296
MSC (2010): Primary 65D30; Secondary 41A55
DOI: https://doi.org/10.1090/S0025-5718-2010-02386-5
Published electronically: June 7, 2010
MathSciNet review: 2728980
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Abstract: We consider in this paper asymptotic and numerical aspects of highly oscillatory integrals of the form $ \int_a^b f(x) g(\sin[\omega \theta(x)])\mathrm{d} x$, where $ \omega\gg1$. Such integrals occur in the simulation of electronic circuits, but they are also of independent mathematical interest.

The integral is expanded in asymptotic series in inverse powers of $ \omega$. This expansion clarifies the behaviour for large $ \omega$ and also provides a powerful means to design effective computational algorithms. In particular, we introduce and analyse Filon-type methods for this integral.


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Additional Information

Arieh Iserles
Affiliation: Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
Email: ai@damtp.cam.ac.uk

David Levin
Affiliation: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
Email: levin@tau.ac.il

DOI: https://doi.org/10.1090/S0025-5718-2010-02386-5
Received by editor(s): October 30, 2008
Received by editor(s) in revised form: August 21, 2009
Published electronically: June 7, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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