Paley-Wiener type theorems for distributions?

Question

In general a theorem of Paley-Wiener type gives a relation between the decay of a function and the smoothness of its Fourier transformation, and there are plenty of them since there are many kinds of bound for decay rates of functions and many types of characterizations of smoothness.

However,when the objects are tempered distributions I only know one such theorem, the one given in Wiki page as Schwartz's Paley-Wiener theorem, which deals only with the case of compact support distributions.

I wonder whether there are other Paley-Wiener type theorems for distributions that might be less restrictive.

Thanks!

Answer 1

There is also a theorem on distributions of exponential decay.

It states that $f(x) \exp \langle -\lambda,x \rangle$ is tempered for all $\lambda$ in an open convex set $C \ni 0$ iff its Fourier transform $\hat f$ has analytic continuation to $\mathbb{R^n} + iC$, and $\hat f(\cdot + it)$ is of moderate growth uniformly in $t$ on compact subsets of $C$.

I can't give a precise reference, but something like that may be found in the second volume of Reed & Simon's 'Mathematical phisics'.

Answer 2

A standard reference is Hörmander book volume I.