The classical Shannon sampling theorem states that a bandlimited function with $\mbox{supp } \hat f\subset [-1/2,1/2]$ can be uniquely determined by its samples $(f(i))_{i\in \mathbb{Z}}$ (The symbol $\hat f$ refers to the Fourier transform of $f$).

My question is as follows: Suppose that $f$ is not bandlimited but that we have an estimate of the form

$$ |\hat f (\omega)|\le C \exp(-|\omega|^2). $$

Can such $f$ still be determined by its samples at the integers?

Edit: The answer to the question as stated is `NO', by the example of Jean Duchon. But is the following true: Suppose that $\hat f$ satisfies the above growth estimate. Can $f$ be uniquely determined by its samples $(f(\alpha i))_{i\in\mathbb{Z}}$ for all $\alpha <1$?

some$\alpha<1$? $\endgroup$ – Sebastian Goette Mar 3 '16 at 16:04