Is there a version of Paley-Wiener theorem for functions of finite order? 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.
 A: The following argument is similar to the classical calculation of the order in terms of the Taylor coefficients.
Statement. Let $\Omega_{\delta} = \{z\colon \text{Im}(z) > -\delta\}$ and $\mathbb{R}_{\delta} = \{z\colon \text{Im}(z) = -\delta\}$. Assume that the entire function $F$ satisfies the following:

*

*$F(z)\le C_1e^{c_2|\text{Im}(z)|^{\beta}}$ for some $c_1,c_2$  and $\beta > 1$;

*$F\in H^2(\Omega_{\delta})$ for all $\delta > 0$;

*$\|F\|_{L^1(\mathbb{R}_{\delta})} \le C_1e^{c_2\delta^{\beta}}$  for all $\delta > 0$.

Then there exists $\phi\in L^2(\mathbb{R}_+)$ and $\alpha = \frac{\beta}{\beta - 1}$ such that $|\phi(x)|\le c_3e^{c_4|x|^{-\alpha}}$ and $F(z) = \int_0^{\infty}\phi(x)e^{izx}dx $ for all $z\in\mathbb{C}$.
Proof: since $F\in H^2(\mathbb{C}_+)$ there is some $\phi\in L^2(\mathbb{R}_+)$ satisfying $F(z) = \int_0^{\infty}\phi(x)e^{izx}dx $ for all $z\in\mathbb{C}_+$. Similarly for each $\delta > 0$ there exists $\phi_{\delta}\in L^2(\mathbb{R}_+)$ such that $F(z - i{\delta}) = \int_0^{\infty}\phi_{\delta}(x)e^{izx}dx$. It follows that
$$
F(z) = \int_0^{\infty}\phi_{\delta}(x)e^{i(z - ic)x}dx = \int_0^{\infty}\phi_{\delta}(x)e^{-cx}e^{izx}dx,\qquad \phi(x) = \phi_{\delta}(x)e^{-\delta x}.
$$
Next, we have
$$
\phi(x)e^{cx} = \phi_{\delta}(x) = \frac{1}{2\pi}\int_{\mathbb{R}} F(\lambda - i\delta)e^{-\lambda x}d\lambda,
\\
|\phi(x)e^{\delta x}|\le C \|F\|_{L^1(\mathbb{R}_{\delta})} \le C'e^{c_2{\delta}^{\beta}}.
$$
For all $\delta > 0$ we get the bound
$$
|\phi(x)|\le c\exp(c\delta^{\beta} - \delta x),
$$
which is optimal when $\delta \approx x^{1/(\beta  -1)}$ and this implies $|\phi(x)|\le ce^{cx^{\beta/(\beta - 1)}}$.
