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.