A Family of Smooth Quasi-interpolants Defined Over Powell–Sabin Triangulations

Abstract

We investigate the construction of local quasi-interpolation schemes based on a family of bivariate spline functions with smoothness \(r\ge 1\) and polynomial degree \(3r-1\). These splines are defined on triangulations with Powell–Sabin refinement, and they can be represented in terms of locally supported basis functions that form a convex partition of unity. With the aid of the blossoming technique, we first derive a Marsden-like identity representing polynomials of degree \(3r-1\) in such a spline form. Then we present a general recipe to construct various families of smooth quasi-interpolation schemes involving values and/or derivatives of a given function.