In this talk we shall present a few recent results concerning the approximation of functions in Banach spaces of distributions (or just measurable functions) by finite linear combinations of {\it translations of a given functions}.

The key feature of the approach is the use of integrated group actions (realized as convolution with elements from a Beurling algebra or and pointwise multiplication with respect to some Fourier-Beurling algebra), the use of bounded approximate units, and discretization of such convolution products.

These results have been obtained in joint work with Anupam Gumber (the first paper is about to appear in Proc. AMS soon).

In a second paper (preprint available on ARXIV) we show how these results
can be improved: using a Tauberian condition a single function (without dilations) can be used.

As a by-product (making use of the characterization of compact sets in those Banach spaces, which are double Banach modules) it can be demonstrated that all those (separable) Banach spaces of tempered distributions satisfy \tblue{ the bounded approximation property}. Such a statement had been shown already in a widely unknown paper by the author (jointly with W.Braun), published in 1985, using fairly abstract methods involving twisted convolution.