On spectral expansions of piecewise smooth functions depending on the geodesic distance

We consider the expansion of a piecewise smooth function depending on the geodesic distance to some point in the eigenfunctions of the Beltrami-Laplace operator on an n-dimensional symmetric space of rank 1. We show that if the expansion converges at this point, then the function must have continuous derivatives up to and including the order (n − 3)/2.