Quasi-interpolants splines : exemples et applications

Free access article

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


Page(s)		195 - 207
DOI		http://dx.doi.org/10.1051/proc:072017
Published online		13 October 2007

ESAIM: Proc., 2007, Vol. 20, pp. 195-207
DOI: 10.1051/proc:072017

Quasi-interpolants splines : exemples et applications

Paul Sablonnière

Paul Sablonnière, Centre de mathématiques, INSA de Rennes, 20 avenue des Buttes de Coësmes, CS 14315, 35043 Rennes cédex, France. e-mail : psablonn@insa-rennes.fr

(Published online: 13 October 2007)

Abstract
A quasi-interpolant (abbr. QI) is an approximation operator obtained as a finite linear combination of basis functions with bounded support (B-splines). In addition, the coefficient of a B-spline is a linear functional (of differential, discrete or integral type) acting on the function to be approximated in a neighbourhood of the support of that B-spline. The big advantage of this approach is that the computation of a QI is direct and does not need the solution of any system of equations, unlike what happens with interpolants. It is particularly interesting in the bivariate or trivariate cases, where the number of B-splines can be rather large. In this paper, I present some examples of QIs of different types on spaces of univariate or multivariate splines. Then, I give some applications to approximation and numerical analysis.

Résumé
Un quasi-interpolant spline (abréviation QI) est un opérateur d'approximation obtenu comme combinaison linéaire de fonctions à support borné (B-splines) :

$\begin{displaymath}Qf=\sum_{\alpha\in A} \mu_\alpha(f) B_\alpha\end{displaymath}$

Le coefficient $\mu_\alpha(f)$ de la B-spline $B_\alpha$ est une forme linéaire agissant sur la fonction f à approcher dans un voisinage du support de $B_\alpha$ . Le grand avantage de cette approche est que le calcul d'un QI est direct et ne nécessite pas la résolution d'un système d'équations, contrairement à ce qui se passe pour un interpolant. C'est particulièrement intéressant en dimension 2 ou 3, où le nombre de B-splines peut être relativement grand. Dans cet article, je décris quelques exemples de QIs de différents types sur des espaces de splines à une ou deux variables. Puis je présente quelques applications en approximation et en analyse numérique.

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