How well does the Hermite-Padé approximation smooth the Gibbs phenomenon?

Authors:
Bernhard Beckermann, Valeriy Kalyagin, Ana C. Matos and Franck Wielonsky

Journal:
Math. Comp. **80** (2011), 931-958

MSC (2010):
Primary 41A21, 41A20, 41A28, 42A16, 31C15, 31C20

Published electronically:
September 27, 2010

MathSciNet review:
2772102

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Abstract | References | Similar Articles | Additional Information

Abstract: In order to reduce the Gibbs phenomenon exhibited by the partial Fourier sums of a periodic function , defined on , discontinuous at 0,

Driscoll and Fornberg considered so-called singular Fourier-Padé approximants constructed from the Hermite-Padé approximants of the system of functions , where and is analytic, such that . Convincing numerical experiments have been obtained by these authors, but no error estimates have been proven so far. In the present paper we study the special case of Nikishin systems and their Hermite-Padé approximants, both theoretically and numerically. We obtain rates of convergence by using orthogonality properties of the functions involved along with results from logarithmic potential theory. In particular, we address the question of how to choose the degrees of the approximants, by considering diagonal and row sequences, as well as linear Hermite-Padé approximants. Our theoretical findings and numerical experiments confirm that these Hermite-Padé approximants are more efficient than the more elementary Padé approximants, particularly around the discontinuity of the goal function .

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Additional Information

**Bernhard Beckermann**

Affiliation:
Laboratoire Mathématiques, P. Painlevé UMR CNRS 8524, Université de Lille 1, France

Email:
bbecker@math.univ-lille1.fr

**Valeriy Kalyagin**

Affiliation:
Higher School of Economics Nizhny Novgorod, Russia

Email:
kalia@hse.nnov.ru

**Ana C. Matos**

Affiliation:
Laboratoire Mathématiques, P. Painlevé UMR CNRS 8524, Université de Lille 1, France

Email:
Ana.Matos@math.univ-lille1.fr

**Franck Wielonsky**

Affiliation:
Laboratoire Mathématiques, P. Painlevé UMR CNRS 8524, Université de Lille 1, France

Email:
Franck.Wielonsky@math.univ-lille1.fr

DOI:
https://doi.org/10.1090/S0025-5718-2010-02411-1

Received by editor(s):
July 7, 2009

Received by editor(s) in revised form:
January 15, 2010

Published electronically:
September 27, 2010

Additional Notes:
This work was supported by INTAS network NeCCA 03-51-6637 and partly by RFBR 08-01-00179

Article copyright:
© Copyright 2010
American Mathematical Society