On the convergence rate of a modified Fourier series

Author:
Sheehan Olver

Journal:
Math. Comp. **78** (2009), 1629-1645

MSC (2000):
Primary 42A20

DOI:
https://doi.org/10.1090/S0025-5718-09-02204-2

Published electronically:
February 18, 2009

MathSciNet review:
2501067

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Abstract | References | Similar Articles | Additional Information

Abstract: The rate of convergence for an orthogonal series that is a minor modification of the Fourier series is proved. This series converges pointwise at a faster rate than the Fourier series for nonperiodic functions. We present the error as an asymptotic expansion, where the lowest term in this expansion is of asymptotic order two. Subtracting out the terms from this expansion allows us to increase the order of convergence, though the terms of this expansion depend on derivatives. Alternatively, we can employ extrapolation methods which achieve higher convergence rates using only the coefficients of the series. We also present a method for the efficient computation of the coefficients in the series.

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Additional Information

**Sheehan Olver**

Affiliation:
Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford, United Kingdom

Email:
sheehan.olver@sjc.ox.ac.uk

DOI:
https://doi.org/10.1090/S0025-5718-09-02204-2

Keywords:
Orthogonal series,
function approximation,
oscillatory quadrature.

Received by editor(s):
April 22, 2008

Published electronically:
February 18, 2009

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.