On the construction of trivariate near-best quasi-interpolants based on C2 quartic splines on type-6 tetrahedral partitions

On spline quasi-interpolation through dimensions

“…Finally, the third one, Q 3 , is constructed so that it is exact on cubic polynomials and in addition it shows 2. Moreover, the construction of new QIs based on the same trivariate C 2 quartic box spline, having optimal approximation order and small infinity norm is addressed in [3]. Such near-best QIs are obtained imposing exactness on the space of cubic polynomials and minimizing an upper bound of their infinity norm which depends on a finite number of free parameters.…”

“…In particular, the aim of this paper is to collect the main results obtained by the authors, also in collaboration with other researchers, in such a topic, highlighting the approximation properties and the reconstruction of functions and data [3, 5, 6, 8-10, 26, 32-34, 36, 43, 45, 47, 57, 58, 67, 69-71], the applications in numerical integration and differentiation [22, 25, 27-31, 35, 37, 39-42, 44, 56, 68] and the numerical solution of integral and differential problems [2,13,24,38,46]. The above results can also be considered through spline dimension: 1D [2, 13, 25, 28, 29, 35-37, 39-42, 46, 57, 68], 2D [6, 8, 9, 22, 24, 26, 27, 30-34, 38, 43, 45, 56-58, 67, 70] and 3D [3,5,10,44,47,69,71].…”

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.…”