Reduction of the Gibbs phenomenon for smooth functions with jumps by the ε-algorithm

Authors:
Bernhard Beckermann;Ana C. Matos;Franck Wielonsky
Affiliations:
Laboratoire Painlevé UMR 8524, UST Lille, F-59655 Villeneuve d'Ascq CEDEX, France;Laboratoire Painlevé UMR 8524, UST Lille, F-59655 Villeneuve d'Ascq CEDEX, France;Laboratoire Painlevé UMR 8524, UST Lille, F-59655 Villeneuve d'Ascq CEDEX, France
Venue:
Journal of Computational and Applied Mathematics
Year:
2008

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Abstract

Recently, Brezinski has proposed to use Wynn's @e-algorithm in order to reduce the Gibbs phenomenon for partial Fourier sums of smooth functions with jumps, by displaying very convincing numerical experiments. In the present paper we derive analytic estimates for the error corresponding to a particular class of hypergeometric functions, and obtain the rate of column convergence for such functions, possibly perturbed by another sufficiently differentiable function. We also analyze the connection to Pade-Fourier and Pade-Chebyshev approximants, including those recently studied by Kaber and Maday.