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IMA Journal of Numerical Analysis Advance Access published online on April 2, 2008
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn006
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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 I: modified Fourier expansions

Arieh Iserles{dagger}

DAMTP, 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: a.iserles{at}damtp.cam.ac.uk

Received on 20 June 2006. Revised on 31 December 2007.


  Abstract

In this paper, we consider a modification of the classical Fourier expansion, whereby in [– 1, 1] the sin {pi}nx functions are replaced by sin {pi}(n –1/2)x, n ≥ 1. This has a number of important advantages in the approximation of analytic, nonperiodic functions. In particular, expansion coefficients decay like Formula (n–2), rather than like Formula (n–1). We explore theoretical features of these modified Fourier expansions, prove suitable versions of the Fejér and de la Vallée Poussin theorems and expand the coefficients into asymptotic series. This expansion is a key towards the computation of expansion coefficients by asymptotic and Filon-type methods. We explore this issue in some detail and present a number of algorithms which require Formula (m) operations in the computation of the first m expansion coefficients.

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


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