The classical Paley-Wiener theorem says that any entire function of exponential type $\sigma$ which is square-integrable over real line is the Fourier transform of an $L^2$ function supported in $[−\sigma, -\sigma]$. Is it possible to similarly describe functions of finite order? I did not find any results of this kind in the literature and will be grateful for suggestions of references.
My thought is that it can be connected to the superexponential decay of the Fourier transform. For example if the function $g$ defined on $\mathbb{R}$ satisfies the assertion $|g(x)|\le e^{-|x|^{\alpha}}$ for some $\alpha > 1$ then $$ \left|\mathcal{F}(g)(\lambda)\right| = \left|\int_{\mathbb{R}}g(x)e^{i\lambda x}dx \right|\le \int_{\mathbb{R}}e^{-x^{\alpha} + |x\Im \lambda}|dx \le C \exp\left(|\Im \lambda|^{\frac{\alpha}{\alpha - 1}}\right), $$ hence $\mathcal{F}(g)$ defines an entire function of order not greater $\frac{\alpha}{\alpha - 1}$. This extends to the situation when $\int_{r}^{r + 1} |g(x)|\,dx\le c_1e^{-c_2|r|^{\alpha}}$ for some sonstants $c_1$ and $c_2$. However the inverse does not seem to be true.