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IMA Journal of Numerical Analysis Advance Access originally published online on July 25, 2008
IMA Journal of Numerical Analysis 2009 29(4):882-916; doi:10.1093/imanum/drn020
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

From high oscillation to rapid approximation III: multivariate expansions

Arieh Iserles{dagger}

Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK

Syvert P. Nørsett

Department of Mathematics, Norwegian University of Science and Technology, Trondheim N-7491, Norway

{dagger} Email: ai{at}damtp.cam.ac.uk

Received on 15 February 2007. Revised on 26 February 2008.


  Abstract

In this paper, we expand upon the theme of modified Fourier expansions and extend the theory to a multivariate setting and to expansions in eigenfunctions of the Laplace–Neumann operator. We pay detailed attention to expansions in a d-dimensional cube and to an effective derivation of expansion coefficients there by means of quadratures of highly oscillatory integrals. Thus, we present asymptotic and Filon-type formulae for an effective derivation of expansion coefficients and discuss their design and relative advantages. Such methods are effective only for large indices; hence, we introduce and analyse alternative quadrature schemes that require a relatively modest number of additional function evaluations.

Key Words: Fourier expansion; highly oscillatory quadrature; Filon-type methods


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