Wavelet Based Solutions to the Poisson and the Helmholtz Equations on the n-Dimensional Unit Sphere

Abstract

We present a method of solving partial differential equations on the n-dimensional unit sphere using methods based on the continuous wavelet transform derived from approximate identities. We give an explicit analytical solution to the Poisson equation and to the Helmholtz equations. For the first one and for some special values of the parameter in the latter one, we derive a closed formula for the generalized Green function.