My question is whether Dirac-type distributions over an Abelian group define a basis of the Schwartz-Bruhat space $\mathcal{S}(G)^\times$ of tempered distributions on $G$, so that any distribution $f\in\mathcal{S}(G)^\times$ can be expressed as an integral of Dirac deltas
$$f=\int\_X \mathrm{d}\mu(x)\,\, f(x)\, \delta\_x $$
for some subset $X$ of $G$, some measure $\mu$ on $X$ and some function $f:X\rightarrow \mathbb{C}$.


I have attempted to solve this myself without much success. (I am a quantum physicist and lack background in harmonic analysis.) You can take a look at my **attempt** and a at more *precise* version of my **question** below.




---


Let $G$ be a Hausdorff locally compact Abelian group, $\mathcal{H}=L^2(G)$ the Hilbert space of two-integrable functions with its usual inner product. Let $\mathcal{S}(G)$ be the [Schwartz-Bruhat space](http://en.wikipedia.org/wiki/Schwartz%E2%80%93Bruhat_function) [ [2](http://archive.numdam.org/ARCHIVE/BSMF/BSMF_1961__89_/BSMF_1961__89__43_0/BSMF_1961__89__43_0.pdf), [3](http://www.sciencedirect.com/science/article/pii/0022123675900051) ] of smooth functions of rapid decay over $G$ and $\mathcal{S}(G)^\times$ its continuous dual vector space of **tempered distributions**, which I want to view it as a $\mathbb{C}$-vector space. The triple


$$\mathcal{S}(G)\subset \mathcal{H}\subset \mathcal{S}(G)^\times$$


is a rigged Hilbert space.


**Question 1**. One of the properties of the dual $\mathcal{S}(G)^\times$ is that it contains all Dirac-type delta distributions $\delta\_g$ defined as 


$$\langle \delta\_g , \varphi\rangle = \varphi(g).$$


I would like to understand for which groups the set $\{\delta\_g,\, g\in G\}$ defines a basis in $\mathcal{S}(G)^\times$. I assume already that $\mathcal{H}$ has a basis. The easy example is when $G$ is a finite Abelian group: in this case the $\delta\_g$ represent Kronecker deltas and it can be shown that they define a basis of the dual space $\mathcal{S}(G)^\times$ using linear algebra. 


**Question 2. (A particular case.)** If the answer to the above questions is negative I would be very interested to know what happens for groups of the form $$G=\mathbb{R}^a\times\mathbb{Z}^b\times \mathbb{R/Z}^c\times F$$ where $F$ is finite Abelian. This is actually the class of groups I am working with on a project. These groups are compactly-generated LCA groups, and are sometimes easier to work with.




---


**My attempt.** Since every tempered distribution $T$ can be obtained as the limit of a sequence $\{f\_n\}\_n$ of Schwartz-Bruhat functions, my (possibly wrong) intuition that I $\{\delta\_g\}$ should define a basis of $\mathcal{S}(G)^\times$, because one should be able to write every function of the sequence as a linear combination of deltas *exclusively*


$$ f\_n = \int\_{G} \mathrm{d} g \,\, f\_n(g)\,\,\delta\_g$$


(where we integrate over the Haar measure of $G$), I think this should imply that $f$ is a linear combination of deltas in the limit, due to the fact that in the limit


$$\lim\_{n\rightarrow \infty}\langle f\_n , \varphi \rangle= \langle f, \varphi \rangle$$


for every test function $\varphi \in \mathcal{S}(G)$. But honestly, I do not know whether this that this argument is 100% correct and neither I know whether one can formalize it. I wonder also whether this type of argument could work for the class of groups $G=\mathbb{R}^a\times\mathbb{Z}^b\times \mathbb{R/Z}^c\times F$.



Is every distribution a linear combination of Dirac deltas?
The short answer is that the $\delta\_x$'s alone are not enough for a "basis". But it is enough to simply include all possible derivatives $\partial\delta\_x$, $\partial\partial \delta\_x$, ..., as well.


