Help with proof of Poisson summation formula

Question

I am trying to understand a proof of the Poisson summation formula and I cannot understand a vital part of it which the author seems to think is obvious, but is not obvious to me. If anyone can fill me in on the details of this step I would be very, very obliged:

If $f \in \mathcal{S(\mathbb{R})}$, and if we let $F(x) := \sum_{k=-\infty}^\infty f(x+k)$, we then have that $F(x) = \sum_{k=-\infty}^\infty \hat{F}e^{ikx}$.

Thanks

Answer 1

Note that $F(x)$ is periodic with period $1$ (from the first representation in your question), so it can be represented by a Fourier series, where $\hat{F}_k$ are its Fourier coefficients (the second representation in your question). As it turns out later in the proof, the coefficients $\hat{F}_k$ are equidistant samples of the Fourier transform of $f(x)$.