Quasi-interpolation operators based on the trivariate seven-direction C 2 quartic box spline

Abstract

This paper investigates the space S (X) generated by the integer translates of the trivariate C 2 quartic box spline B defined by a set X of seven directions that forms a regular partition of the space into tetrahedra.In S (X) local spline quasi-interpolants are defined by linear combinations of integer translates of B {B α , α ∈ Z 3 } and local linear functionals {D f (α), α ∈ Z 3 }. First a quasi-interpolant of differential type is proposed and its coefficient functionals are linear combinations of values of f with its partial derivatives at the center of the support of B α . Then, by convenient discretisations of the above differential coefficients, three quasi-interpolants of discrete type with different properties are defined. In this case the coefficient functionals are linear combinations of values of f at specific points in the support of B α .Upper bounds both for the norm of the discrete quasi-interpolation operators and for the approximation error are given.Finally, some numerical examples, illustrating the approximation properties of the proposed quasi-interpolants, are presented.

NURBS on Criss-cross Triangulations and Applications

“…Such spaces are defined on complex 3D partitions, as for example tetrahedral or prismatic ones (see e.g. [7,26,33,34,35,36]). …”

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

“…1] and [11,Chap. 15,16,17]). Its smoothness depends on the determination of the number d, such that d + 1 is the minimal number of directions to be removed from X to obtain a reduced set, that does not span R 3 .…”

“…3 (b), (c), (d)) can be thought of as points outside Ω and projected on ∂Ω. We also notice that recently, in [16], QIs based on the same box spline B have been proposed, but they are defined on the whole space R 3 . Therefore, if only a finite number of volume data is available, such schemes can reconstruct only a portion of the volumetric object.…”

Numerical integration based on trivariate C 2 quartic spline quasi-interpolants

“…In the space S 2 4 (Ω , T m ), we consider several quasi-interpolation operators [28]. A quasi-interpolant is a linear operator defined on a functional space F , in the following way…”

“…it is constructed by minimizing an upper bound of its infinity norm. The fourth one Q 4 is exact on P 3 and shows some superconvergence properties at specific points of the domain (the vertices and the centers of each cube of the partition, see [28] for more details).…”