Characterisation of Fourier transform of compactly supported measure

Question

I am aware of the well-known Paley-Wiener theorem stating that the Fourier transform is a bijection from the set of compactly supported distributions to the space of exponentially bounded entire functions. I am looking for a reference if there is a similarly nice characterisation of compactly supported Radon measures on $\mathbb{R}$. In other words, when is an entire function the Fourier transform of a compactly supported measure? Ultimately, I am interested in when a set of numbers in $\mathbb{C}$ are the roots of the Fourier transform of a nonzero compactly supported Radon measure, so any reference on this would also be greatly appreciated!