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

Abstract

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 quasiinterpolants, some numerical examples are presented and compared with those obtained by some other trivariate quasi-interpolants given recently in the literature.

Near-best $$C^2$$ C 2 quartic spline quasi-interpolants on type-6 tetrahedral partitions of bounded domains

“…Remogna and Sablonnière [18] propose a near-best quasi-interpolant, with approximation order three, defined as blending sum of univariate and bivariate C 1 quadratic spline quasi-interpolants, on a partition of Ω into vertical prisms with triangular sections. We denote it by R 1 .…”

“…A possible 3D spline model, beyond the classical tensor product schemes, is represented by blending sums of univariate and bivariate C 1 quadratic spline quasi-interpolants (see e.g. [18,19,20]), defined on a bounded domain, partitioned into vertical prisms with triangular horizontal sections. In this case the approximation order is three, as for triquadratic splines, but the degree of the piecewise polynomials is only four, instead of six.…”

Weighted Quasi-Interpolant Spline Approximations of Planar Curvilinear Profiles in Digital Images

“…A rather simple generalization, known as Variation Diminishing Spline Approximation (VDSA), generalizes this construction to B-splines (see, for example [5,6]). Since its inception, quasi-interpolation has been studied to obtain methods that apply to different domains and with the aim of increasing the order of convergence: recent developments include univariate and tensorproduct spaces [7][8][9], triangular meshes [10][11][12][13], quadrangulations [14] and tetrahedra partitions [15], among others.…”

Multivariate normalized Powell–Sabin B-splines and quasi-interpolants

“…Bivariate quadratic Powell-Sabin quasi-interpolants have been developed in [10,18], and trivariate Powell-Sabin quasi-interpolants in [19]. There exist many other quadratic quasi-interpolants, for example in two variables [3,13] and in three variables [11,14,15].…”