Spherical Singular Integrals, Monogenic Kernels and Wavelets on the Three-Dimensional Sphere

Abstract

The construction of wavelets relies on translations and dilations which are perfectly given in \({\mathbb{R}}\). On the sphere translations can be considered as rotations but it is difficult to say what are dilations. For the 2-dimensional sphere there exist two different approaches to obtain wavelets which are worth to be considered. The first concept goes back to W. Freeden and collaborators who define wavelets by means of kernels of spherical singular integrals. The other concept developed by J.P. Antoine and P. Vandergheynst is a purely group theoretical approach and defines dilations as dilations in the tangent plane. Surprisingly both concepts coincides for zonal functions. We will define singular integrals and kernels of singular integrals on the three dimensional sphere which are also approximate identities. In particular the Cauchy kernel in Clifford analysis is a kernel of a singular integral, the singular Cauchy integral, and an approximate identity. Furthermore, we will define wavelets on the 3-dimensional sphere by means of kernels of singular integrals.