I have the following problem:
If a Schwartz funcetion $f\in\mathcal{S}(\mathbb{R})$ satisfies $f(2\pi n)=0$ and $\hat{f}(n)=0,\forall n\in\mathbb{Z}$, where $\hat{f}(\xi)=\int e^{-ix\xi}f(x)dx$ is the Fourier transform of $f$. Prove that $f=0$.
If $f$ is compactly supported, then Fourier series gives the result. However, I don't know how to deal with the Schwartz function. Could anyone solve it? Thanks a lot!