Let $G$ be a Hausdorff locally-compact **Abelian** group and $L^2(G)$ the Hilbert space of two-integrable complex functions on the group.

Question.What would be naturalvector space$\mathcal{R}$ of complex functions such that its dual vector space $\mathcal{R}^\times$ of continuous complex linear functionals fulfill the following properties?:

(1)$\mathcal{R}^\times$ contains all Dirac delta measures of the group, i.e. the functionals $$\langle \delta_g , \varphi\rangle = \varphi(g),\quad \textrm{ for all }\, g\in G,\,\varphi\in\mathcal{R}.$$

(2)Any functional in $f\in\mathcal{R}^\times$ can be expressed as some integral of Dirac deltas $$f=\int_X \textrm{d}\mu(x)\,\, f(x)\, \delta_x $$ for some subset $X$ of $G$, some (suitable) measure $\mu$ on $X$ and some (suitable) complex function $f$.

(3)In (2) there is flexibility to choose the allowed classes of measures $\mu$ and complex functions $f$ as long as $$\mathcal{R}\subset L^2(G)\subset \mathcal{R}^\times$$ is a rigged Hilbert space.

**Motivation**

A space $\mathcal{R}$ chosen as above would be a *minimal rigged Hilbert space*$\mathcal{R}\subset L^2(G)\subset \mathcal{R}^\times$ that is *wide enough* to contain the Dirac measures as distributions but *tight* in the sense that any other object in $\mathcal{R}^\times$ can be written as a "linear combination" (an integral as in property 1.) of Dirac deltas. In this case, one can think of the set $\{\delta_g\}$ as an (infinite) *basis* of the vector space $\mathcal{R}^\times$. My original motivation was to understand for what kind of distribution spaces (over groups) the Dirac deltas $\{\delta_g\}$ define such a "basis". (See also this previous post of mine.)

**Can $\mathcal{R}$ have nice dual properties?**

Since $G$ is an LCA group we can consider its dual group of continuous character functions $\widehat{G}$; then Plancherel's theorem tells us that $L^2(G)$ is isomoprhic to $L^2(\widehat{G})$ via the Fourier transform of $G$. For applications in harmonic analysis, it would be desirable that the properties (1-3) of $\mathcal{R}^\times$ defined above were preserved under the application of the Fourier transform. In particular, I wonder whether $\mathcal{R}$ could have a 4th additional property. (Perhaps it follows easily from the previous ones?)

(4)$\mathcal{R}^\times$ contains all Dirac delta measures of thedual group$\widehat{G}$ (the group continuous character functions of $G$). In other words, the functionals $$\langle \delta_{\chi} , \varphi\rangle = \langle \chi, \varphi \rangle = \widehat{\varphi}(\chi),\quad \textrm{ for all }\, \chi\in \widehat{G},\,\varphi\in\mathcal{R}.$$

Here, $\mathcal{\varphi}$ denotes the Fourier transform of $\varphi$, which is a function in $L^2(\widehat{G})$