If I understand correctly, Paley–Wiener theorem says that if a function $F:\mathbb{R}\to \mathbb{C}$ is compactly supported, its holomorphic Fourier transform is entire. Just wonder, whether this argument could be extended to generalized function as well, for example, the Dirac delta function
$ \delta(x)=\left\{ \begin{array}{ll} +\infty, & x=0 \\ 0, & x\neq 0. \end{array} \right. $
For this special case, it seems to work because the holomorphic Fourier transform is constantly 1. I am just wondering if there is any general result on this?
Thanks!
If $T$ is a distribution compactly supported on $[a,b]$ then for any $\varphi \in C^\infty_c $ which is $=1$ on $[a-\epsilon,b+\epsilon]$ $$F(z) = \langle T, e^{zx} \rangle =\langle T , e^{zx}\varphi \rangle=\sum_{k=0}^\infty \frac{z^k}{k!} \langle T , x^k\varphi \rangle$$ and hence $F$ is entire.
More generally the same holds if $T$ is a distribution such that $T \ast \phi(x)$ decreases faster than every exponential.