Locally supported wavelets on manifolds with applications to the 2D sphere

In this paper we present a construction principle for locally supported wavelets on manifolds once a multiresolution analysis is given. The wavelets provide a stable (or unconditional) basis for a scale of Sobolev spaces H-s, 0 less than or equal to s less than or equal to (s) over bar. We examine a fast wavelet transform with almost optimal complexity. For the two-dimensional sphere we construct a multiresolution analysis generated by continuous splines that are bilinear with respect to some special spherical grid. In our approach the poles are not exceptional points concerning the approximation power or the stability of the wavelet basis. Finally we present some numerical applications to singularity detection and the analysis of observed atmospheric data. (C) 1999 Academic Press.