On spline quasi-interpolation in cubic spline space S 31,2(Δ mn (2))

In this paper, by means of the basis composed of two sets of splines with distinct local supports, cubic spline quasi-interpolating operators are investigated on nonuniform type-2 triangulation. These variation diminishing operators based on five mesh points or the center of the support of each spline B ij 1 and five mesh points of the support of each spline B ij 2 can preserve good approximation, and even reproduce any polynomial of nearly best degrees. Moreover, the spline series can approximate a real sufficiently smooth function uniformly based on the modulus of continuity. And then the convergence results are worked out.