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

DOI:
https://doi.org/10.1090/S0025-5718-2010-02393-2

Published electronically:
August 13, 2010

MathSciNet review:
2728978

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Abstract | References | Similar Articles | Additional Information

Abstract: Modified Fourier series have recently been introduced as an adjustment of classical Fourier series for the approximation of nonperiodic functions defined on -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

DOI:
https://doi.org/10.1090/S0025-5718-2010-02393-2

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.