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 $\alpha <1$?



Sampling Theorem for non-bandlimited Functions
No as well to the edited question (if I understood it...): $f(x)=e^{-cx^2}\sin(\pi x/\alpha)$


