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The main result of this paper is a far reaching generalization of thecompleteness result given by V.~Katsnelson in a recent paper [35]. Instead ofjust using a collection of dilated Gaussians it is shown that the key steps ofan earlier paper [27] by the authors, combined with the use of Tauberianconditions (i.e. the non-vanishing of the Fourier transform) allow us to showthat the linear span of the translates of a single function $g \in{{{\boldsymbol{\mathcal {S}}}({\mathbb{R}}^d)}}$ is a dense subspace of anyBanach 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 $({\boldsymbol B}, \, \|\,\cdot\,\|_{\boldsymbolB})$ one can construct a finite rank operator, whose range is contained in thelinear span of finitely many translates of $g$, and which approximates theidentity operator over $M$ up to a given level of precision.  The setting of tempered distributions allows to reduce the technicalarguments to methods which are widely used in Fourier Analysis. The extensionto non-quasi-analytic weights respectively locally compact Abelian groups isleft to a forthcoming paper, which will be technically much more involved anduses different ingredients.

Completeness of Sets of Shifts in Invariant Banach Spaces of Tempered Distributions via Tauberian conditions