Multiresolution analysis over triangles, based on quadratic Hermite interpolation

Given a triangulation T of R2, a recipe to build a spline space S(T) over this triangulation, and a recipe to refine the triangulation T into a triangulation T′, the question arises whether S(T)⊂S(T′), i.e., whether any spline surface over the original triangulation T can also be represented as a spline surface over the refined triangulation T′. In this paper we will discuss how to construct such a nested sequence of spaces based on Powell–Sabin 6-splits for a regular triangulation. The resulting spline space consists of piecewise C1-quadratics, and refinement is obtained by subdividing every triangle into four subtriangles at the edge midpoints. We develop explicit formulas for wavelet transformations based on quadratic Hermite interpolation, and give a stability result with respect to a natural norm.