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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.


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up vote 4 down vote accepted

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'.

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Once one knows an approximately correct statement, it is not hard to prove such a thing as an exercise, I think. – paul garrett Oct 3 '12 at 14:50
I'm not quite sure that I understand the statement. What does "$\hat{f}$ has an analytic continuation to $\mathbb{R}^n+iC$" mean if $f$ is a distribution? Does this mean that "$\hat{f}$ is a continuous/smooth function on $\mathbb{R}^n$" is part of the claim? – Johannes Hahn Apr 26 '14 at 17:30
@JohannesHahn: Yes, in particular $\hat f$ happens to be a smooth function. The usual intuition works: smoothness of $\hat f$ is related to the decay of $f$ at infinity. – Alexander Shamov Apr 26 '14 at 23:48

A standard reference is Hörmander book volume I.

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