Let $$f(x) = \sum_{n \in \mathbb{Z}} \hat{f}(n) e^{2 \pi i nx},$$ be the Fourier series for $f \in L^2[0,1]$. If $f \neq 0$ almost surely, show that $\{\hat{f}(n+m)\}_{m \in \mathbb{Z}}$ is complete in $\ell^2(\mathbb{Z})$. In other words the closed span of $\{\hat{f}(n+m)\}_{m \in \mathbb{Z}}$ is dense in $\ell^2(\mathbb{Z})$. Let $\{\hat{g}(n)\}_{n \in \mathbb{Z}}$ be such that $\langle \hat{g}(n), \hat{f}(n+m)\rangle = 0$ for all $m \in \mathbb{Z}$. Then, $$0 = \langle \hat{g}(n), \hat{f}(n+m)\rangle = \langle g(x), \sum_n \hat{f}(n+m)e^{2 \pi i nx} \rangle = \sum_n \hat{f}(n+m)\langle g(x),e^{2 \pi inx} \rangle.$$ I have not used the fact that $f \neq 0$ so maybe that should come into play here? I am not sure how to proceed. I know that if $\langle h(x),e^{2 \pi i nx}\rangle = 0$ for all $n$, then we have $h = 0$, so maybe I need to use this here. Please help.
Thanks :)!