Superconvergent C1 cubic spline quasi-interpolants on Powell-Sabin partitions

In this paper we introduce a B-spline representation of the cubic Hermite Powell-Sabin interpolant of any polynomial or any piecewise polynomial, over Powell-Sabin partitions of class at least , in terms of their polar forms. We use this B-spline representation for constructing several superconvergent discrete cubic spline quasi-interpolants which approximate a function better than the superconvergent quadratic ones developed in one of our recent published papers. The new results presented in this work are an improvement and a generalization of those studied recently in the literature. We also illustrate by numerical examples that global errors and cubature rules based on these cubic Powell-Sabin spline quasi-interpolants are positively affected by the superconvergence phenomenon.