**"Approximation by Translates in Translation and Modulation Invariant Banach Spaces"**
#### Gumber, AnupamUsing well-established methods from the theory of Banach modules and time-frequency analysis, the main result of present paper provides an alternative approach to the completeness result of a recent paper by V.~Katsnelson (in: Complex Anal. Oper. Theory, 2019). Instead of just using a collection of dilated Gaussians it is shown that the key steps of an earlier paper (in: Proc. Amer. Math. Soc., 2021) by the authors, combined with the use of some conditions (i.e. the non-vanishing of the Fourier transform) allow us to show in a constructive manner that the linear span of the translates of a single Schwartz function $g$ over $R^d$ is a dense subspace of any Banach space $B$ satisfying certain double invariance properties. In fact, a much stronger statement is presented: for a given compact subset $M$ in such a Banach space $B$ 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. There is a comprehensive list of examples of such spaces, and they do not have to be contained in $L^2(R^d)$. This is a joint work with Hans G. Feichtinger. |