Polar forms and quadratic spline quasi-interpolants on Powell–Sabin partitions

In this paper we introduce the Powell–Sabin B-spline representation of quadratic polynomials or splines in terms of their polar forms. We use this B-representation for constructing several differential or discrete quasi-interpolants which have an optimal approximation order. This new approach is simple and provides an efficient tool for describing many schemes of approximation involving values and (or) derivatives of a given function.