Let $f\in L_{\operatorname{loc}}^2 (\mathbb{R})$ be a function of period $2\pi$. I'd like to get some help proving the following identity:
$$\hat{f}=\sum_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$$
With some basic calculations, it seems that proving the following identity should suffice:
$$\int \int f(x)e^{-ix\xi} \phi(\xi)d\xi dx=\sum_{n=-\infty}^{\infty}\int e^{-inx}f(x)\phi(n)dx\mbox{ for every }\phi \in S(\mathbb{R}).$$
However, it seems that I can't prove this identity either, so perhaps this is not the right way to attack the problem, so I will appreciate any help proving the first identity (whether by proving the second identity or in any other way).