  The main result of this paper is a far reaching generalization of the
completeness result given by V. Katsnelson in a recent paper [28]. Making use
of Tauberian Theorems for Beurling algebras as found in the book of H. Reiter
[31] as well as the theory of invariant Banach spaces of tempered distributions
we demonstrate that instead of using all dilates of a given function with
non-vanishing integral (this case is treated in [22]) it is enough to use one
single function $g$ such as the Gaussian and its translates in order to
generate a dense subspace of the given invariant space. The key condition now
is the non-vanishing of the Fourier transform $\hat{g}(y)$, for any $y \in
\mathbb{R}^d$.
