On the approximation power of bivariate quadratic C1 splines

Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains

“…In [9][10][11] some tools are given to construct a more general class of spline QIs, among which the two above mentioned, defined as a linear combination of such C 1 quadratic box-splines, whose coefficients are in turn linear combinations of values of the function to be approximated. In those papers their effective construction is studied and the corresponding Matlab software is given.…”

“…In [7,8] some approximation power performances and error analysis on the function and on its partial derivatives of first and second order with local and global upper bounds are given for such a general class of spline QIs.…”

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

“…Furthermore, according to classical results in approximation theory (see e.g. DeVore andLorentz 1993, Chap.5, Lyche andSchumaker 1975 for the univariate case and de Boor et al 1993, Chap.3, Dagnino andLamberti 2001;Foucher and Sablonnière 2008;Lyche and Schumaker 1975 for the bivariate one) and, in view of the exactness of P on P p [x, y] andP on Pp[z], we have that the rate of convergence is O(h p ) for the two-dimensional case and O(hp) for the univariate case, i.e.…”

On the Approximation of the Derivatives of Spline Quasi-Interpolation in Cubic Spline $S_3^{1,2}(∆_{mn}^{(2)})$

“…These spline quasi-interpolating operators can reproduce any polynomial of (nearly) best degrees, respectively. Moreover, spline quasiinterpolation defined by discrete linear functionals based on a fixed number of triangular mesh-points has been investigated, which showed that they could approximate a real function and its partial derivatives up to an optimal order in [1,2].…”

“…In Section 2, we review the dimension and the bases in S 1,2 3 (∆ (2) mn ). Based on five mesh points or the center of the support of each spline B 1 i j and five mesh points of the support of each spline B 2 i j , the representation of spline quasi-interpolation is investigated, which can reproduce any polynomial in 2 ∪ {x 2 y, x y 2 }. Then in section 3, we make a further study of the derivatives of the cubic spline quasi-interpolation, which can approximate the derivatives of the real sufficiently smooth function uniformly over quasi-uniform triangulation.…”