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Issue ESAIM: Proc.
Volume 20, 2007
RFMAO 05 - Rencontres Franco-Marocaines en Approximation et Optimisation 2005
Page(s) 29 - 43
DOI http://dx.doi.org/10.1051/proc:072003
Published online 13 October 2007

ESAIM: Proc., 2007, Vol. 20, pp. 29-43
DOI: 10.1051/proc:072003

Approximation by New Families of Univariate Symmetrical B-splines

El Bachir Ameur1, Hamid Mraoui2 and Driss Sbibih1

1  Département d'Informatique, Faculté des Sciences et Techniques, Errachidia, Maroc
2  Université Mohammed I, Ecole Supérieure de Technologie , Laboratoire MATSI, Oujda, Maroc

eb_ameur@yahoo.fr
sbibih@yahoo.fr
hamid_mraoui@yahoo.fr

(Published online: 13 October 2007)

Abstract
In this paper we prove that there exists a unique positive symmetrical univariate B-spline with minimal support. It is obtained as linear combination of a minimal number of successive classical B-splines with multiple knots in the space, $\mathcal{S}^r_d$, of cardinal polynomial splines of class ${\mathcal C}^r$ and degree d. Next, we show that the approximation order in the space generated by the integer translates of this B-spline is not optimal. However it can be used for geometrical design where the small support is appreciated but the approximation order is not crucial. To have a higher approximation order, we define the B-splines of high order by recurrence and by convolution with the characteristic function of the interval [0,1]. We use these B-splines to study the cardinal interpolation and we show that it is correct in the sense of Schoenberg. Finally, we give the explicit expression of interplant operators associated with some of these B-splines.


Résumé
Dans cet article, nous montrons l'existence et l'unicité d'une spline symétrique à support minimal qui s'écrit comme combinaison linéaire d'un nombre minimal de B-spline successives de l'espace $\mathcal{S}^r_d$ des splines polynomiales cardinales de degré d et de régularité r. Nous démontrons que l'ordre d'approximation dans l'espace engendré par les translatés entières de cette B-spline n'est pas optimal. Cependant, leur utilisation dans le dessin géométrique, où l'ordre d'approximation n'est pas crucial mais où un support de longueur réduit est recommondé, pourrait être très utile. Pour avoir un ordre d'approximation élevé, nous définissons par récurrence de nouvelles familles de B-splines cardinales symétriques. Ensuite, nous étudions l'unisolvance du problème d'interpolation basé sur ces B-splines. Nous donnons enfin, des exemples de calcul des coefficients des splines fondamentales associées à quelques éléments de degrés faibles de ces nouvelles familles.


Mathematics Subject Classification. 41A15, 65D05, 65D07, 65D10.

Key words: B-splines, Bm-splines, fundamental splines, cardinal interpolation.


© EDP Sciences, ESAIM 2007


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