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Thread: Poisson summation formula

  1. #1
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    Poisson summation formula

    I just have no idea how to apply Poisson summation formula. Can someone give some help?
    a) Let $\displaystyle \tau$ be fixed with $\displaystyle Im(\tau)>0$. Apply the Poisson summation formula to $\displaystyle f(z)=(\tau +z)^{-k}$ where $\displaystyle k \geq 2$ to obtain
    $\displaystyle \sum_{n=-\infty}^{\infty} \frac {1}{(\tau+n)^k}= \frac{(-2\pi i)^k}{(k-1)!} \sum_{m=1}^{\infty} m^{k-1} e^{2 \pi im \tau}$
    b) Does this formula still hold whenever $\displaystyle \tau$ is any complex number that is not an integer?
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  2. #2
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    Quote Originally Posted by namelessguy View Post
    I just have no idea how to apply Poisson summation formula. Can someone give some help?
    a) Let $\displaystyle \tau$ be fixed with $\displaystyle Im(\tau)>0$. Apply the Poisson summation formula to $\displaystyle f(z)=(\tau +z)^{-k}$ where $\displaystyle k \geq 2$ to obtain
    $\displaystyle \sum_{n=-\infty}^{\infty} \frac {1}{(\tau+n)^k}= \frac{(-2\pi i)^k}{(k-1)!} \sum_{m=1}^{\infty} m^{k-1} e^{2 \pi im \tau}$
    b) Does this formula still hold whenever $\displaystyle \tau$ is any complex number that is not an integer?
    As for how to apply Poisson sommation formula, everything is given in the text: they even give the function to use. The formula says:
    $\displaystyle \sum_{n\in\mathbb{Z}} f(n)=\sum_{n\in\mathbb{Z}} \widehat{f}(n)$,
    provided the second sum is absolutely convergent. And if the Fourier transform is defined as $\displaystyle \widehat{f}(\xi)=\int e^{-2i\pi x\xi}f(x)dx$.

    The left-hand side is exactly what you want, so you have to check that the right-hand side matches the formula you're given. In other words, you have to compute the Fourier transform of $\displaystyle f$.

    This can be tedious, but there are tricks:
    first, notice that $\displaystyle f(x)=\frac{(-1)^{k-1}}{(k-1)!}\frac{d^{k-1}}{dx^{k-1}}\left(\frac{1}{\tau+x}\right)$, hence $\displaystyle \widehat{f}(\xi)=\frac{(-1)^{k-1}}{(k-1)!}(2i\pi\xi)^{k-1}\mathcal{F}\left(\frac{1}{\tau+x}\right)(\xi)$. You have to be careful here, since $\displaystyle x\mapsto \frac{1}{\tau+x}$ is not integrable, but only square-integrable.
    Then you can notice that (by easy integration) $\displaystyle \mathcal{F}^{-1}\left(H(\xi)e^{2i\pi \xi\tau}\right)(x)=\frac{1}{-2i\pi(\tau+x)}$ where $\displaystyle H$ is the Heavyside function ($\displaystyle H(x)=1$ if $\displaystyle x\geq 0$ and $\displaystyle H(x)= 0$ else). As a consequence, using Fourier inversion formula (in $\displaystyle L^2$), $\displaystyle \mathcal{F}\left(\frac{1}{\tau+x}\right)(\xi)=(-2i\pi)H(\xi)e^{2i\pi \xi\tau}$.
    We deduce $\displaystyle \widehat{f}(\xi)=\frac{(-1)^{k-1}}{(k-1)!}(2i\pi\xi)^{k-1}(-2i\pi)e^{2i\pi\xi\tau}H(\xi)$ in $\displaystyle L^2$ (hence for almost every $\displaystyle \xi$), and hence for every $\displaystyle \xi$ because both functions are continuous (even for $\displaystyle \xi=0$ because $\displaystyle k\geq 2$).

    As a consequence, $\displaystyle \widehat{f}(n)=\frac{(-2i\pi)^k}{(k-1)!}n^{k-1}e^{2i\pi n\tau}$ if $\displaystyle n> 0$ and 0 otherwise.

    The series $\displaystyle \sum_n \widehat{f}(n)$ is absolutely convergent because $\displaystyle |e^{2i\pi n\tau}|=e^{{\rm Re}(2i\pi n\tau)}=e^{-2\pi n {\rm Im}(\tau)}$ and $\displaystyle {\rm Im}(\tau)>0$. That's why we can apply the Poisson sommation formula. And it gives exactly what is expected.

    By the way, if $\displaystyle {\rm Im}(\tau)\leq 0$, the series would diverge.
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  3. #3
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    Thanks a lot for your help Laurent. I really need to review this Fourier Transform materials. I wasn't able to apply this concept to show that $\displaystyle \sum_{n=-\infty}^{\infty} \frac {1}{(\tau+n)^2}=\frac{\pi^2}{sin^2(\pi \tau)}$. I plug in $\displaystyle k=2$ on both sides of the formula, but got nowhere.
    I then tried an alternative way suggested by my friend. I integrate $\displaystyle f(z)=\frac{\pi cot \pi z}{(\tau + n)^2}$ over the circle $\displaystyle \mid z \mid=R_N$. I almost got the result (still working on showing $\displaystyle cot \pi z$ is bounded in that circle).
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  4. #4
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    Quote Originally Posted by namelessguy View Post
    Thanks a lot for your help Laurent. I really need to review this Fourier Transform materials. I wasn't able to apply this concept to show that $\displaystyle \sum_{n=-\infty}^{\infty} \frac {1}{(\tau+n)^2}=\frac{\pi^2}{sin^2(\pi \tau)}$. I plug in $\displaystyle k=2$ on both sides of the formula, but got nowhere.
    For this, you should compute $\displaystyle \sum_{m=1}^\infty m x^m$ where $\displaystyle x=e^{2i\pi\tau}$. This sum is obtained by differentiating $\displaystyle \sum_{m=0}^\infty x^m=\frac{1}{1-x}$.
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