Journal of computational and applied mathematics vol:170 issue:1 pages:181-196
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 L-P norm. Here the approximation constants depend also on a fraction proper to the triangulation, thus the B-splines are not stable for the L-P 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.