  The aim of this exposition is to explain basic ideas behind the concept of
diffusive wavelets on spheres in the language of representation theory of Lie
groups and within the framework of the group Fourier transform given by
Peter-Weyl decomposition of $L^2(G)$ for a compact Lie group $G$.
  After developing a general concept for compact groups and their homogeneous
spaces we give concrete examples for tori -which reflect the situation on
$R^n$- and for spheres $S^2$ and $S^3$.
