Let X be a symmetric
space and G the connected component of the group of isometries of X. If
f ∈ L1(X), we consider conditions under which Sp{gf : g ∈ G} is dense in L1(X)
in terms of the “Fourier transform” of f. This continues earlier work on
this kind of problem by L. Ehrenpreis and F. I. Mautner, R. Krier and the
author.