On trivariate blending sums of univariate and bivariate quadratic spline quasi-interpolants on bounded domains

• Published in: Computer aided geometric design Vol. 28; no. 2; pp. 89 - 101
• Main Authors: Remogna, S., Sablonnière, P.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 01-02-2011; Elsevier
• Subjects: Applied sciences; Approximation; Blending; Boundaries; Computer aided design; Computer science; control theory; systems; Exact sciences and technology; Functionals; Generators; Mathematical analysis; Mechanical engineering. Machine design; Norms; Quasi-interpolation operator; Software; Spline approximation; Splines; Trivariate splines; Quasi-interpolation operator; Trivariate splines; Spline approximation; Multivariate interpolation; Error estimation; Best approximation; Computer aided design
• ISSN: 0167-8396; 1879-2332
• DOI: 10.1016/j.cagd.2010.12.002

The aim of this paper is to investigate, in a bounded domain of
R
3
, two blending sums of univariate and bivariate
C
1
quadratic spline quasi-interpolants.
The main problem consists in constructing the coefficient functionals associated with boundary generators, i.e. generators with supports not entirely inside the domain. In their definition, these functionals involve data points lying inside or on the boundary of the domain. Moreover, the weights of these functionals must be chosen so that the quasi-interpolants have the best approximation order and a reasonable infinite norm.
We give their explicit constructions, infinite norms and error estimates. In order to illustrate the approximation properties of the proposed quasi-interpolants, some numerical examples are presented and compared with those obtained by some other trivariate quasi-interpolants given recently in the literature.
► Spline approximation. ► Quasi-interpolation operator. ► Trivariate splines.