Bivariate C 2 cubic spline quasi-interpolants on uniform Powell–Sabin triangulations of a rectangular domain

Abstract

In this paper we construct discrete quasi-interpolants based on C 2 cubic multi-box splines on uniform Powell-Sabin triangulations of a rectangular domain. The main problem consists in finding the coefficient functionals associated with boundary multi-box splines (i.e. multi-box splines whose supports overlap with the domain) involving data points inside or on the boundary of the domain and giving the optimal approximation order. They are obtained either by minimizing an upper bound for the infinity norm of the operator w.r.t. a finite number of free parameters, or by inducing the superconvergence of the gradient of the quasi-interpolant at some specific points of the domain.Finally, we give norm and error estimates and we provide some numerical examples illustrating the approximation properties of the proposed operators.

On spline quasi-interpolation through dimensions

“…In the second case, we impose superconvergence of the operator at some specific points of : we require that the quasi-interpolation error at such points is O(h 4 ), beside a global error O(h 3 ). In [70] the same approaches are used in the space of C 2 cubic splines on uniform Powell-Sabin triangulations of a rectangular domain.…”

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.…”

Construction of a normalized basis of a univariate quadratic $C^1$ spline space and application to the quasi-interpolation

“…The coefficients of the linear combination are the values of linear functionals, depending on f and (or) its derivatives or integrals. Many works concerning the construction of quasi-interpolant are developed in the literature (see [1,2,3,8,4,5,14,13,21,22,23]). The main gain of these operators is that they have a direct construction without solving any system of equations and with the minimum possible computation time.…”