Electronic Transactions on Numerical Analysis vol:25 pages:224-258
Recently we developed multiscale spaces of C^1 piecewise quadratic polynomials on the Powell-Sabin 6-split of a triangulation relative to arbitrary polygonal domains Omega subset of R^2. These multiscale bases are weakly stable with respect to the L_2 norm. In this paper we prove that these multiscale spaces form a multiresolution analysis for the Banach space C^1(Omega) and we show that the multiscale basis forms a strongly stable Riesz basis for the Sobolev spaces H^s(Omega) with s is an element of (1, 5/2). In other words, the norm of a function f is an element of H^s(Omega) can be determined from the size of the coefficients in the multiscale representation of f. This property makes the multiscale basis suitable for surface compression. A simple algorithm for compression is proposed and we give an optimal a priori error bound that depends on the smoothness of the input surface and on the number of terms in the compressed approximant.