I'm currently trying to figure out necessary conditions on functions which would be solutions of the problem in the post Functions $f(x)$ for which the set of functions $af(x+b)$ is closed under addition. If I'm correct (whether I'm correct is a separate issue...), then a uniformly continuous solution would induce a (tempered) distribution whose Fourier transform has finite support (not just compact!).
However I'm quite new to the theory of distributions as a whole and especially to the theory of Fourier transforms of tempered distributions, and so I don't really have an idea as to how restrictive this condition would actually be.
My question today will thus be very simple:
Is there a known characterisation of distributions whose Fourier transform has finite support?
I did hear/read about the Paley-Wiener theorem characterising distributions whose Fourier transform is compactly supported, so at least I know that such distributions are actually functions and that they're entire with some growth restrictions, so there's already a lot of regularity at play. Yet, outside of that, I didn't really see much of anything about my particular concern, though to be fair it's probably because the keywords "finite" and "discrete" when searched alongside "fourier transform" seem typically tied to matters like "Fourier transform whose support lies in a finite measure subset", "discrete Fourier transform", and the likes, thus there could be a post somewhere about exactly my question but it's drowned in the sea of more popular queries so to speak?
My current guess as of writing this question would be that a distribution with finite support is a linear combination of terms of the form $\delta^{(k_j)}(\xi - a_j)$ where $\delta$ is the Dirac delta, as I did read somewhere (can't find it again though) that a distribution with support in $\{0\}$ are linear combinations of derivatives of $\delta$, but I could be completely wrong. If that's correct, then the inverse Fourier transform would be of the type $\sum_{j} \lambda_j x^{k_j}e^{ia_j x}$, right? This would be very simple if that's the case, but I'm not confident on that and I don't know the details that go into the claims I assumed.
(If you have a good starting reference about distributions and tempered distributions, do feel free to share it if you'd want to. I'd like to learn more about distributions and techniques tied to distributions. But you don't have to if you'd just rather answer the question, no problem.)
Feel free to re-tag and edit if/when necessary or appropriate.
Thanks to Jochen's kind comment, I found pretty much what I wanted in Rudin's Functional Analysis. Since I did forget in the question to specify that I was in $\mathbb{R}$: I'll write what happens for $\mathbb{R}$, but rest assured Rudin covers general $\mathbb{R}^d$.
Theorem $6.25$ of Rudin's FA indeed states that distributions of support $\{p\}$ (of finite order $N$ technically, but the previous Theorem $6.24(d)$ establishes that compactly supported distributions have a finite order) are necessarily finite linear combinations of $\delta_p$ and its partial derivatives. It takes a bit to write down but it essentially comes down to proving that such a distribution vanishes on the intersection of the kernels of a finite number of the derivatives of the Dirac delta at $p$, hence the result by an elementary lemma of linear algebra (Lemma $3.9$ in Rudin).
Now, the characterisation of distributions with finite support is left as one of the exercises for that chapter, but I think I've figured it out.
Consider a set $\{p_1, \dots, p_r\} \subset \mathbb{R}^d$ ($p_k \neq p_l$ for $k \neq l$). Let $\Lambda$ be our distribution supported in $\{p_1, \dots, p_r\}$. Define $\varepsilon := \frac{1}{2}\min_{k \neq l} |p_k - p_l| > 0$, and $V_k := B_{\mathbb{R}, |\cdot|}(p_k, \varepsilon)$. Finally, take $(\psi_k)_k$ any $\mathcal{C}^\infty$-partition of unity subordinate to $(V_1, \dots, V_r)$ (for example the one from Theorem $6.20$).
Then, we have $\Lambda = \sum_{k = 1}^r \psi_k\Lambda$, where each $\psi_k\Lambda$ is a distribution supported in $V_k \cap \{p_1, \dots, p_r\} = \{p_k\}$, hence each $\psi_k\Lambda$ is a finite linear combination of Dirac delta derivatives at $p_k$, which implies that $\Lambda$ is a finite linear combination of Dirac delta derivatives at the points $p_k$.
Lastly, let's see what the inverse Fourier transform of $\delta_{p}^{(k)}$ yields: differentiation under the inverse Fourier transform is equivalent to multiplying said inverse Fourier transform by $-ix$ and vice-versa (see the Tempered Distributions part of Rudin's FA for the correct formalism for general $\mathbb{R}^d$), hence we only need to look at what $\mathcal{F}^{-1}[\delta_p]$ is, and: $$\begin{split}\left\langle \mathcal{F}^{-1}[\delta_p], \varphi\right\rangle \overset{\text{by def.}}{=} \left\langle \delta_{p}, \mathcal{F}^{-1}[\varphi]\right\rangle &= \mathcal{F}^{-1}[\varphi](p)\\ &= \frac{1}{\sqrt{2\pi}}\int_\mathbb{R} e^{ipx} \varphi(x) \mathrm{d}x\\ &= \left\langle \frac{1}{\sqrt{2\pi}}e^{ipx}, \varphi\right\rangle \end{split}$$ (I have doubts about the constants involved but they're only constants thankfully...) This all means that what I said was correct and distributions whose Fourier transform have finite support are exactly the functions of the type $\sum_j \lambda_j x^{k_j} e^{ia_jx} $, but now it should be (hopefully!) backed up with actual proofs.