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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: 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.