How Does the Poisson Summation Formula Apply to Uniformly Convergent Series?

[mr.tea]

Homework Statement

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 $u(x)= \sum_{-\infty}^{\infty} g(x+2n\pi)$

Homework Equations

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.

 

[mighty2000]

Dear student,

Thank you for your question. The Poisson summation formula is a powerful tool in mathematical analysis and has wide applications in various fields such as physics, engineering, and statistics. It relates the values of a function at integer points to the values of its Fourier coefficients. In this case, we are given 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 $0\leq x \leq 2\pi$. This means that the functions $g(x+2n\pi)$ and $g'(x+2n\pi)$ are well-behaved and do not have any large fluctuations or oscillations in this interval.

Now, the hint given in the problem suggests that we consider the function $u(x) = \sum_{-\infty}^{\infty} g(x+2n\pi)$, which is a $2\pi$-periodic function. This means that $u(x+2\pi) = u(x)$ for all values of $x$. We can also express this function in terms of its Fourier series as:

$u(x) = \sum_{m=-\infty}^{\infty} \gamma_m e^{imx}$

where the coefficients $\gamma_m$ are given by $\gamma_m = \frac{1}{2\pi} \int_{-\pi}^{\pi} u(x) e^{-imx} dx$.

Now, let us consider the function $f(x) = \sum_{-\infty}^{\infty} g(2n\pi) e^{inx}$, which is also a $2\pi$-periodic function. We can write this as:

##f(x) = \sum_{n=-\infty}^{\infty} g(2n\pi) e^{inx} = \sum_{n=-\infty}^{\infty} \left(\sum_{m=-\infty}^{\infty} \gamma_m e^{im(2n\pi)}\right) e^{inx} = \sum_{n=-\infty}^