Wiener's Tauberian Theorem 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 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, 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.

*Korevaar, Jacob*, **Tauberian theory. A century of developments**, Grundlehren der Mathematischen Wissenschaften 329. Berlin: Springer (ISBN 3-540-21058-X/hbk). xvi, 483 p. (2004). ZBL1056.40002.