Wavelets on Manifolds I: Construction and Domain Decomposition

The potential of wavelets as a discretization tool for the numerical treatment of operator equations hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet expansion coefficients and on certain cancellation properties. These features are crucial for the construction of optimal preconditioners, for matrix compression based on sparse representations of functions and operators as well as for the design and analysis of adaptive solvers. So far the availability of such bases is confined to very simple domain geometries. This paper is concerned with concepts that aim at expanding the applicability of wavelet schemes. The central issue is to construct wavelet bases with the desired properties on manifolds which can be represented as the disjoint union of smooth parametric images of the standard cube. The approach is based on the characterization of function spaces over such a manifold in terms of product spaces where each factor ..