[Wiener's Tauberian Theorem][1] says that linear combinations of translations of a function $f$ are dense in $L^1(\mathbb{R})$ if and only if the zero set of the Fourier transform of $f$ is empty. This is an old theorem, from 1932(ish). I am interested in generalizations such as: 

 - When can a nonnegative $L^1$ function approximated by _nonnegative_ linear combinations of translations of $f\ge 0$? 
 - What about convex combinations for approximating a density function?
 - Other spaces: What about $L^1(X)$ or $L^p(X)$ for $X\ne\mathbb{R}$, $p\ne1,2$? ([This question][2] is also related.)

I am looking for any modern reference covering these types of extensions, or other interesting extensions of this theorem. 

**Edit:** I eventually found [this tome][3], which covers Tauberian theory in great detail, but not necessarily the approximation problems above. At 500 pages, it will take me some time to go through it.

<cite authors="Korevaar, Jacob">_Korevaar, Jacob_, [**Tauberian theory. A century of developments**](http://dx.doi.org/10.1007/978-3-662-10225-1), Grundlehren der Mathematischen Wissenschaften 329. Berlin: Springer (ISBN 3-540-21058-X/hbk). xvi, 483&nbsp;p. (2004). [ZBL1056.40002](https://zbmath.org/?q=an:1056.40002).</cite>


  [1]: https://en.wikipedia.org/wiki/Wiener%27s_tauberian_theorem
  [2]: https://mathoverflow.net/questions/4216/is-there-an-lp-tauberian-theorem?rq=1
  [3]: https://www.springer.com/gp/book/9783540210580