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Poisson summation formula

  1. Oct 5, 2016 #1
    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. jcsd
  3. Oct 10, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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