A B-spline-like basis for the Powell-Sabin 12-split based on simplex splines (Q2840627)

Using a single macroelement coming from the celebrated Powell-Sabin split, the authors study piecewise polynomial spline surfaces over triangulations (two dimensions). Their piecewise polynomial degree is two. The generated bases are unconditionally stable bases so that B-spline type basis functions for the triangulations and the aforementioned spline spaces arise. They are continuously differentiable, and the authors derive all the essential properties and formulae in this very well readable article, namely recurrence relations for differentiation and pointwise function evaluation, Marsden identities, and suitable smoothness conditions. Also, two further highly relevant approaches to functional approximation are studied, namely subdivision and the important and useful quasi-interpolation.