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?

Answers or references to learn about those questions are very welcome.

Note

I noticed the existence of these Schwartz-Bruhat functions, which seem to be related to my questions. If the theory of these is the key to the answer, a good reference would also be welcome.

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