# Poisson summation formula

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1. Oct 5, 2016

### mr.tea

1. The problem statement, all variables and given/known data
let $g$ be a $C^1$ function such that the two series $\sum_{-\infty}^{\infty} g(x+2n\pi)$ and $\sum_{n=-\infty}^{\infty} g'(x+2n\pi)$ are uniformly convergent in the interval $0\leq x \leq 2\pi$. Show the Poisson summation formula:

$\sum_{n=-\infty}^{\infty} g(2n\pi) = \sum_{-\infty}^{\infty} \gamma _m$

where $\gamma _m= \frac{1}{2\pi} \int_{-\infty}^{\infty} g(x)e^{-imx} dx$ is assumed to be convergent.
Hint: The numbers $\gamma _m$ are the Fourier coefficients of the $2\pi$-periodic function $u(x)= \sum_{-\infty}^{\infty} g(x+2n\pi)$

2. Relevant equations

3. The attempt at a solution
I have tried to use the hint, but arrived nowhere. Also, I am not sure why do I need the differentiated series $\sum_{-\infty}^{\infty} g'(x+2n\pi)$ ...

Thank you.

2. Oct 10, 2016