This is a follow-up on my last question about distributions, Distributions whose Fourier transforms have finite support, again to potentially help with figuring out the solutions for the post Functions $f(x)$ for which the set of functions $af(x+b)$ is closed under addition.
To have this be self-contained: last time I managed, thanks to a kind commenter who gave me a few references, to show that tempered distributions whose Fourier transforms have finite support are exactly the functions of the form $$x \in \mathbb{R} \mapsto \sum_{j = 1}^r \lambda_j x^{k_j} e^{ia_j x} \in \mathbb{C}$$ with $\lambda_j \in \mathbb{C}$, $a_j \in \mathbb{R}$ and $k_j \in \mathbb{N}$.
The result follows from Rudin's Functional Analysis for the case of a single point and then a straightforward partition of unity argument to combine the information given by the Fourier transform at each point of its support to obtain that it is a finite sum of derivatives of the Dirac delta at said points.
My question today is about what happens if instead of finite support, we consider countable and discrete support?
The motivation is that the Fourier transform of the solution to the latter post is itself a solution of the equation $g(\xi)\hat{f}(\xi) = 0$ in the distributional sense on $\mathbb{R}$ for some function $g$ extendable to a holomorphic function on $\mathbb{C}$.
Thus the support of $\hat{f}$ is contained in the set of zeroes of $g$, which is discrete and countable by the identity theorem/principle of isolated zeroes. The finiteness was obtained with an additional assumption on $f$, however I would like to remove that assumption and deal with a more general case.
Now, onto the question for today: I can still use a locally finite partition of unity instead of a finite one thanks to discreteness, which means that we can write, at least formally: $$\hat{f}(\xi) = \sum_{j \in J} \mu_j \, \delta_{a_j}^{(k_j)}(\xi)$$
with $J$ a countable, possibly infinite indexing set and with the sum being well-defined since it contains only finitely many non-zero terms for each point $\xi$.
Moreover, for each point $a$ in the support of $\hat{f}$, only finitely many $a_j$s can be equal to $a$, and, since a tempered distribution is a distribution of finite order, I'm fairly sure the $k_j$s have to be bounded from above (otherwise you could just pick a sequence of compactly supported functions $(\varphi_{j_n})_n$ which would go be a "witness" for a subsequence $k_{j_n} \to \infty$).
What can be done now? This seems well-defined as a distribution, since testing against compactly supported functions means, with discreteness, only having to deal with finitely many points $a$ of the support at any given time and thus only finitely many terms of the sum above.
However, which conditions are necessary and/or sufficient to ensure this defines a tempered distribution, which we could then apply the inverse Fourier transform to? And what would $f$ look like?
Feel free to edit and re-tag if appropriate or necessary.