Wavelet Approximations on Closed Surfaces and Their Application to Boundary-value Problems of Potential Theory

Wavelets on closed surfaces in Euclidean space R 3 are introduced starting from a scale discrete wavelet transform for potentials harmonic down to a spherical boundary. Essential tools for approximation are integration formulas relating an integral over the sphere to suitable linear combinations of function values (resp. normal derivatives) on the closed surface under consideration. A scale discrete version of multiresolution is described for potential functions harmonic outside the closed surface and regular at innnity. Furthermore, an exact fully discrete wavelet approximation is developed in case of band-limited wavelets. Finally, the role of wavelets is discussed in three problems, namely (i) the representation of a function on a closed surface from discretely given data, (ii) the (discrete) solution of the exterior Dirichlet problem, and (iii) the (discrete) solution of the exterior Neumann problem. 1 The support by "Stiftung Rheinland-Pfalz f ur Innovation" and "Graduiertenkolleg Technomathematik, Kaiserslautern" is greatly acknowledged.