Trivariate near-best blending spline quasi-interpolation operators

Abstract

A method to define trivariate spline quasi-interpolation operators (QIO) is developed by blending univariate and bivariate operators whose linear functionals allow oversampling. In this paper, we construct new operators based on univariate B-splines and bivariate box splines, exact on appropriate spaces of polynomials and having small infinity norms. An upper bound of the infinity norm for a general blending trivariate spline QIO is derived from the Bernstein-Bézier coefficients of the fundamental functions associated with the operators involved in the construction. The minimization of the resulting upper bound is then proposed and the existence of a solution is proved. The quadratic and quartic cases are completely worked out and their exact solutions are explicitly calculated.

On spline quasi-interpolation through dimensions

“…where S 1 and S 1 denote the univariate and bivariate Schoenberg variation-diminishing operator exact on linear polynomials, respectively, while T and T represent univariate and bivariate optimal approximation operators, respectively. In particular, in [5] univariate and bivariate C 1 quadratic spline QIs are considered and a trivariate QI of near-best type is constructed, i.e. the coefficients functionals are determined by minimizing an upper bound of its infinity norm, derived from the Bernstein-Bézier coefficients of its Lebesgue function.…”

“…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].…”