Wavelet analysis and the geometry of Euclidean domains

This paper gives an explicit representation for a multiresolution of Euclidean domains and their boundaries in terms of a wavelet system defined in the ambient space. The exterior derivative of the characteristic function of a domain is represented in an infinite series of compactly supported wavelet functions whose supports intersect the geometric boundary. This is used to obtain representations of the boundary integrals which appear in weak solutions of partial differential equations.