# Distributions over locally compact Abelian groups: when can they be Fourier transformed?

Pontryagin duality shows us every locally compact Abelian group---such as $\mathbb{R}^n$, $\mathbb{Z}$, the circle $\mathbb{R}/\mathbb{Z}$ or any finite Abelian group---has a Fourier transform. In the case of $\mathbb{R}$, it follows from distribution theory that all tempered distributions (including Dirac deltas) have Fourier transforms. I would like to know how these ideas extend to all finite Abelian groups, in particular:

Question 1: how does one define distributions over locally compact Abelian groups?

Question 2: what are the distributions that can be Fourier transformed? what would sets of distributions (for any Abelian group) would be closed under the Fourier transform of the group?