Multiresolution analysis over triangles, based on quadratic Hermite interpolation (Q1576451)

Given a triangulation \(T\) of \( \mathbb R^2\), a recipe to build a spline space \(\mathbb S(T)\) over this triangulation, and a recipe to refine the triangulation \(T\) into a triangulation \(T'\), the question arises whether \(\mathbb S(T)\subset \mathbb S(T')\). In this paper the authors discuss how to construct such a nested sequence of spaced based on Powell-Sabin 6-splits for a regular triangulation. The resulting spline space consists of piecewise \(C^1\)-quadratics, and refinement is obtained by subdividing every triangle into four subtriangles at the edge midpoints. The authors develop explicit formulas for wavelet transformations based on quadratic Hermite interpolation, and give a stability result with respect to a natural norm.