It is not so hard to prove the Nyquist-Shannon theorem. It basically says a real valued function $f$ having a compactly supported Fourier transform function $\hat f$ can be interpolated precisely from the function values at the equally spaced intervals end points. Can we do the same from the function values at the arbitrary uneven interval end points given that the end points are countably infinite?
I suppose this is related to the "rigidity" of the function $f$. When $\hat f$ is compactly supported, the Paley-Wiener theorem states that $f$ can be extended to an entire function on the complex plane bounded by an exponential function. Under this growth restriction, I envision the combination of the Weierstrass and particularly Hadamard's factorization theorems and the Pringsheim interpolation formula would produce the entire function interpolation. I suppose this interpolation would be unique as well. But I so far have not proved it in detail. Could someone please help out?