Pseudo-differential operator associated to the radial basis functions under tension.

Free access article

Issue		ESAIM: Proc. Volume 20, 2007 RFMAO 05 - Rencontres Franco-Marocaines en Approximation et Optimisation 2005


Page(s)		72 - 82
DOI		http://dx.doi.org/10.1051/proc:072007
Published online		13 October 2007

ESAIM: Proc., 2007, Vol. 20, pp. 72-82
DOI: 10.1051/proc:072007

Pseudo-differential operator associated to the radial basis functions under tension.

A. Bouhamidi

L.M.P.A, Université du Littoral Côte d'Opale, 50 rue F. Buisson BP699, F-62228 Calais Cedex, France

A.Bouhamidi@lmpa.univ-littoral.fr

(Published online: 13 October 2007)

Abstract
Radial basis functions under tension (RBFT) depend on a positive parameter, incorporate the concept of spline with tension and provide a convenient way for the control of the behavior of the interpolating surface. The RBFT involve a function which is not complicated than exponential and may be easily coded. In this paper, we show that the RBFT, as like thin plate spline, may be associated to a differential operator in a Beppo-Levi space type. Both smoothing and interpolating problems by RBFT are studied.

Résumé
Les fonctions splines radiales sous tension dépendent d'un paramètre positif. Ces fonctions permettent d'incorporer un concept de tension pour toute dimension de l'espace. On montre dans ce papier que ces fonctions sont associées à un opérateur pseudo-différentiel dans un espace de type Beppo-Levi et on étudie le problème d'interpolation et de lissage. Des examples numériques sont donnés pour illustrer ce type d'approximation.

Mathematics Subject Classification. ???, ???

Key words: Thin plate spline with tension, smoothing and interpolating functions, approximation theory, variational spline, radial basis functions.

© EDP Sciences, ESAIM 2007

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