  The main result of this paper is a far reaching generalization of the
completeness result given by V.~Katsnelson in a recent paper [35]. Instead of
just using a collection of dilated Gaussians it is shown that the key steps of
an earlier paper [27] by the authors, combined with the use of Tauberian
conditions (i.e. the non-vanishing of the Fourier transform) allow us to show
that the linear span of the translates of a single function $g \in
{{{\boldsymbol{\mathcal {S}}}({\mathbb{R}}^d)}}$ is a dense subspace of any
Banach space satisfying certain double invariance properties.
  In fact, a much stronger statement is presented: for a given compact subset
$M$ in such a Banach space $({\boldsymbol B}, \, \|\,\cdot\,\|_{\boldsymbol
B})$ one can construct a finite rank operator, whose range is contained in the
linear span of finitely many translates of $g$, and which approximates the
identity operator over $M$ up to a given level of precision.
  The setting of tempered distributions allows to reduce the technical
arguments to methods which are widely used in Fourier Analysis. The extension
to non-quasi-analytic weights respectively locally compact Abelian groups is
left to a forthcoming paper, which will be technically much more involved and
uses different ingredients.
