f and its fourier transform $\hat{f}$ vanish at integers

Question

If a function f$\in \mathcal{S}$(Schwartz class) satisfies $f(n)=0,\hat{f}(n)=0,n\in \mathcal{Z}$,prove $f=0$.Here $\hat{f}(t)=\int f(x)e^{2\pi itx} dx$.

I want to use Poisson summation fomula.We can derive $\sum_n f(x+n)=0$ and a similar equation for $\hat{f}$ from the condition.But i don't know how to continue.Alternatively,we can consider $f\ast h$ for some particular h(maybe approximations to the identity?) and then use Poission formula,but it doesn't work,either.

Answer 1

Well it is not true that $f$ must vanish identically,

try with any Schwartz function $h$ and

$$f(x) = (e^{2i\pi x}-1) (h(x)-h(x-1))$$