A normalized basis for C1 cubic super spline space on Powell–Sabin triangulation

In this paper, we describe the construction of a suitable normalized B-spline representation for bivariate C1 cubic super splines defined on triangulations with a Powell–Sabin refinement. The basis functions have local supports, they form a convex partition of unity, and every spline is locally controllable by means of control triangles. As application, discrete and differential quasi-interpolants of optimal approximation order are developed and numerical tests for illustrating theoretical results are presented.