Available online at www.sciencedirect.com
Mathematics and Computers in Simulation 77 (2008) 151–160
Near-best operators based on a C
2
quartic spline
on the uniform four-directional mesh
El Bachir Ameur
a
, Domingo Barrera
b
, Mar´ ıa J. Ib´ a˜ nez
b
, Driss Sbibih
c,∗
a
Universit´ e Moulay Ismail, Facult´ e des Sciences et Techniques, D´ epartement d’Informatique, 52000 Errachdia, Morocco
b
Departamento de Matem ´ atica Aplicada, Facultad de Ciencias, Universidad de Granada,
Campus Universitario de Fuentenueva s/n, 18071 Granada, Spain
c
Universit´ e Mohammed I, Ecole Sup´ erieure de Technologie, Laboratoire MATSI, Oujda, Morocco
Available online 31 August 2007
Abstract
We present some results about the construction of quasi-interpolant operators based on a special C
2
quartic B-spline. We show that
these operators, called near-best quasi-interpolants, have the best approximation order and small infinity norms. They are obtained
by solving a minimization problem that admits always a solution. We give an error bound of these quasi-interpolants and we illustrate
our results by a numerical example.
© 2007 IMACS. Published by Elsevier B.V. All rights reserved.
PACS: 41A05; 41A15; 65D05; 65D07
Keywords: B-splines; Box-splines; Subdivision scheme; Refinable function vector; Near-best quasi-interpolants
1. Introduction
Let τ be the uniform triangulation of R
2
whose set of vertices is Z
2
∪ (Z + (1/2))
2
, and whose edges are parallel to
the four directions e
1
= (1, 0), e
2
= (0, 1), e
3
= (1, 1) and e
4
= (-1, 1). Let P
d
be the space of bivariate polynomials
of total degree at most d, and let S
r
d
(τ ) be the space of bivariate piecewise polynomial functions of class C
r
on the plane
and whose restrictions to each triangular cell of τ are in P
d
.
Let T be a triangle of τ and λ = (λ
1
,λ
2
,λ
3
) be the barycentric coordinates of a point M of R
2
relative to T. Each
polynomial p in the space P
d
(T ) of polynomials defined on T has a unique representation in the Bernstein–B´ ezier form
p(M) =
i ∈△
d
b(i) B
d
i
(λ),
Research supported in part by PROTARS III, D11/18, Ministerio de Educacin y Ciencia (Research project MTM2005-01403) and Junta de
Andaluca (research group FQM/191).
∗
Corresponding author.
E-mail addresses: ameurelbachir@yahoo.fr (E.B. Ameur), dbarrera@ugr.es (D. Barrera), mibanez@ugr.es (M.J. Ib´ a˜ nez), sbibih@yahoo.fr
(D. Sbibih).
0378-4754/$32.00 © 2007 IMACS. Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.matcom.2007.08.005