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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).

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I'm not exactly how to make sense of $\hat f(n)$ here; $\hat f$ most likely wont even be a function so how can you evaluate it at $n$? Or by $\hat f(n)$ do you mean the $n$th Fourier coefficient ${1\over 2\pi} \int_0^{2\pi} f(x)e^{-inx}\,dx$? The formula is true with $\hat f(n)$ equal to the $n$th Fourier coefficient (but then the notation is inconsistent). –  Nick Strehlke Jun 2 '13 at 21:38

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