On the stability of normalized Powell–Sabin B-splines

In this paper we show that the normalized Powell–Sabin B-splines form a stable basis for the max norm. The approximation constants depend only on the smallest angle in the underlying triangulation. Since the B-splines refer to the size of the Powell–Sabin triangles, we find that small Powell–Sabin triangles yield better approximation constants than big Powell–Sabin triangles. Next, in addition to the max norm, we treat the Lp norm. Here the approximation constants depend also on a fraction proper to the triangulation, thus the B-splines are not stable for the Lp norm. Finally, as a special case, we consider the B-spline bases obtained from Powell–Sabin triangles with minimal area and pay extra attention to the approximation constants for the max norm.