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.

Solution 1:

On the $n$-dimensional Euclidian space $\mathbb{R}^n$, the class of tempered distributions is defined in terms of the Schwartz space $\mathcal{S} (\mathbb{R}^n)$, which is the space of all functions $f \in C^{\infty} (\mathbb{R}^n)$ such that for every pair of multi-indices $\alpha, \beta$ there exists constants $C_{\alpha, \beta} > 0$ satisfying $$ \rho_{\alpha, \beta} (f) := \sup_{x \in \mathbb{R}^n} \bigg| x^{\alpha} \partial^{\beta} f(x) \bigg| \leq C_{\alpha, \beta}. $$

On $\mathcal{S} (\mathbb{R}^n)$, the Fourier transform $\mathcal{F}$ is defined as $$ (\mathcal{F} f)(\xi) = \int_{\mathbb{R}^n} f(x) e^{-2\pi i x \xi} dx, $$ and one can show that $\mathcal{F} f \in \mathcal{S} (\mathbb{R}^n)$ whenever $f \in \mathcal{S}(\mathbb{R}^n)$. In fact, the Fourier transform $\mathcal{F}$ is a homeomorphism from $\mathcal{S}(\mathbb{R}^n)$ onto $\mathcal{S}(\mathbb{R}^n)$.

The class of tempered distributions on $\mathbb{R}^n$ is now defined as the topological dual space $(\mathcal{S}(\mathbb{R}^n))'$, i.e., a tempered distribution is a continuous linear functional on $\mathcal{S}(\mathbb{R}^n)$. The Fourier transform $\mathcal{F} : \mathbb{S}(\mathbb{R}^n) \to \mathbb{S}(\mathbb{R}^n)$ can be canonically extended to a mapping $\mathcal{F} : \mathbb{S}((\mathbb{R}^n))' \to (\mathbb{S}(\mathbb{R}^n))'$.

A generalisation of the Fourier transform to tempered distributions on locally compact Abelian groups $G$ is straightforward whenever there is an appropriate definition of the Schwartz space on $G$. The space $\mathcal{L} (G)$ defined next is such a generalisation:

Let $\mathcal{A}(G)$ be the space of all function $f \in L^{\infty} (G)$ for which there exists a compact set $C(f) \subset G$ such that for every $n \in \mathbb{N}$ there exists a constant $C_n$ such that for each $k \in \mathbb{N}^+$ $$ \| f|_{G \setminus C(f)^k} \|_{\infty} \leq C_n k^{-n}.$$ Denote $$\mathcal{L}(G) := \{ f \in \mathcal{A} (G) \; | \; \mathcal{F} f \in \mathcal{A} (\widehat{G}) \}. $$

The space $\mathcal{L}(G)$ was introduced by Osborne in his paper On the Schwartz–Bruhat space and the Paley-Wiener theorem for locally compact abelian groups, and it is a characterisation of the Schwartz-Bruhat space $\mathcal{S}_b (G)$, which was introduced by Bruhat's Distributions sur un group localement compact et applications sons a l'étude de représentations des groups p-adiques. The original definition of the Schwartz-Bruhat space given by Bruhat is probably a more straightforward generalisation of $\mathcal{S}(\mathbb{R}^n)$ since it uses so-called polynomial differential operators. However, the treatment by Bruhat uses the structure theory for locally compact Abelian groups extensively, which is clearly more involved than needed in order to define $\mathcal{S}_b (\mathbb{R})$ by Osborne's characterisation.

An English text that I would recommend for a treatment of Bruhat's original definition of the Schwartz-Bruhat space is Wawrzy´nczyk's On Tempered Distributions and Bochner-Schwartz Theorem on Arbitrary Locally Compact Abelian Groups.