## Stable multiresolution analysis on triangles for surface compression

### Abstract

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 $\Omega \subset {\bf R}^2$. 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 ${\bf C}$ and we show that the multiscale basis forms a strongly stable Riesz basis for the Sobolev spaces $H^s(\Omega)$ with $s \in (2, \frac{5}{2})$. In other words, the norm of a function $f \in H^s(\Omega)$ 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.

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### Key words

hierarchical bases, Powell–Sabin splines, wavelets, stable approximation by splines, surface compression

### AMS subject classifications

41A15, 65D07, 65T60, 41A63

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