Let $g, h \in L^2(\mathbb R)$ and $\langle g, h \rangle = \int_{\mathbb R} \overline{g(x)} h(x) dx.$ Let $f\colon\mathbb R \to \mathbb C$ be a nice function (so everything below make sense, e.g., if we want apply Poisson summation formula at some point), and put $$f_n(x)= f(x-n), n \in \mathbb Z.$$ Question: Can we expect: $$ \sum_{n\in \mathbb Z} \langle g, f_n\rangle \langle f_n, h \rangle = \sum_{k\in \mathbb Z} \int_{\mathbb R} \overline{\hat{g}(\xi)} \hat{h}(\xi +k) \hat{f}(\xi) \overline{\hat{f}(\xi+k)} d\xi.$$
My attempt: We may apply apply Passeval's identity then we have $\langle g, f_n\rangle =\langle \hat{g},\hat{ f_n}\rangle = \int_{\mathbb R} \overline{\hat{g}(\xi)} \hat{f}(\xi) e^{2\pi i \xi \cdot n} d\xi .$ Thus, we may rewrite $\sum_{n\in \mathbb Z} \langle g, f_n\rangle \langle f_n, h \rangle$ as
$\sum_{n\in \mathbb Z} \left( \int_{\mathbb R} \overline{\hat{g}(\xi)} \hat{f}(\xi) e^{2\pi i \xi \cdot n} d\xi \right) \left( \int_{\mathbb R} \hat{h}(\xi) \overline{ \hat{f}}(\xi) e^{-2\pi i \xi \cdot n} d\xi \right)$. I'm guessing now I have to some how use Poisson summation formula to get the above equation, but I do not know how...
Any suggestions and comments are Wellcome ...