In Lund, John, and Kenneth L. Bowers. Sinc methods for quadrature and differential equations. SIAM, 1992. is recalled a Paley-Wiener theorem, as follows:
Assume that f is entire and $f \in L^2(\mathbb R)$. If there are positive constant K so that for all $z\in\mathbb C$ $$|f(z)|\leq K e^{\pi |z|},$$ then $\mathcal F(f)\in L^2(-\pi,\pi)$ and $$f(z)=\frac{1}{2\pi}\int_{-\pi}^\pi \mathcal F(f)(x) e^{-ixz}dx.$$ where $\mathcal F(f)$ is the Fourier transform of $f$.
This theorem is stated without proof. I would get an answer that contains this proof, or a reference (online or not) where we can find it.
Thanks.