Plancherel theorem
It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space "X"; it also gives the direct integral decomposition into irreducible representations of the regular representation on L2("X"). In the case of