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Convergence acceleration of modified Fourier series in one or more dimensions

Author: Ben Adcock
Journal: Math. Comp. 80 (2011), 225-261
MSC (2010): Primary 42A20; Secondary 65B99
Published electronically: August 13, 2010
MathSciNet review: 2728978
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Abstract: Modified Fourier series have recently been introduced as an adjustment of classical Fourier series for the approximation of nonperiodic functions defined on $ d$-variate cubes. Such approximations offer a number of advantages, including uniform convergence. However, like Fourier series, the rate of convergence is typically slow.

In this paper we extend Eckhoff's method to the convergence acceleration of multivariate modified Fourier series. By suitable augmentation of the approximation basis we demonstrate how to increase the convergence rate to an arbitrary algebraic order. Moreover, we illustrate how numerical stability of the method can be improved by utilising appropriate auxiliary functions.

In the univariate setting it is known that Eckhoff's method exhibits an auto-correction phenomenon. We extend this result to the multivariate case. Finally, we demonstrate how a significant reduction in the number of approximation coefficients can be achieved by using a hyperbolic cross index set.

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Additional Information

Ben Adcock
Affiliation: DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

Received by editor(s): September 29, 2008
Received by editor(s) in revised form: May 19, 2009
Published electronically: August 13, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.