In particular if I have an array of numbers say, $\{c_m\}_{m\in\mathbb{Z}^n}$. Under what conditions can we say that these are the Fourier coefficients of a distribution?
[For examples Bessel's inequality ($\sum_{m\in\mathbb{Z}^n}|c_m|^2<\infty$) tells us these that these are the Fourier coefficients of an $L^2$ function.]
If the Fourier coefficients are of polynomial growth in the index $n$, then they are the Fourier coefficients of a distribution.
In one variable, we know in various ways that for $|c_n|\le |n|^{-2}$ (or even weaker conditions) that the resulting Fourier series gives a $C^o$ function. Thus, differentiating termwise (which is legitimate, distributionally) produces distributions... with coefficients growing polynomially.
Conversely, every sequence of polynomial-growth coefficients is obtained in such fashion.
(An easy on-line reference is the "Functions on circles" notes on my Functional Analysis course page, referring to 2012-13 notes, at http://www.math.umn.edu/~garrett/m/fun/ )