A weighted Paley–Wiener characterization for entire functions

Let $\mathrm{w}(\omega)$ be a non-negative even weight function, and let $f(z)$ be an entire function. Suppose there is a function $F(\omega)$ such that

$$\int_{-\infty}^{\infty}|F(\omega)|^2\mathrm{w}(\omega)\,d\omega<\infty$$

and, for every $z\in\mathbb{C}$,

$$f(z)=\int_{-\infty}^{\infty}F(\omega)\mathrm{w}(\omega)e^{\mathrm{i}\omega z}\,d\omega.$$

Weighted Paley–Wiener conjecture. The following equivalences hold:

1. $f(z)$ satisfies $|f(z)|<Ce^{L|z|}$ for some $L,C>0$ if and only if the weight function $\mathrm{w}(\omega)$ in the displayed representation can be chosen to have compact support.
2. For an integer $\kappa>1$, $f(z)$ satisfies $|f(z)|<Ce^{L|z|^\kappa}$ for some $L,C>0$ if and only if the weight function can be chosen to satisfy

$$\mathrm{w}(\omega)\leq e^{-B|\omega|^{-\frac{\kappa}{\kappa-1}}}$$

for some $B>0$ and every $\omega\in\mathbb{R}$.