# Interpretation of Poisson Summation Formula

This question arises from a Fourier transform class I took about a year back.

The poisson summation formula is:

$$\displaystyle \sum_{n= - \infty}^{\infty} f(n) = \displaystyle \sum_{k= - \infty}^{\infty} \hat{f}(k)$$

where $\hat{f}$ is the Fourier Transform of $f(x)$.

It is interesting since this is true for all $f(x)$ for which we can define Fourier transform.

Is there a nice (probably physical) interpretation for this?

I am wondering if this characterizes some property which is invariant, some sort of conservation.

For instance, if we consider Parseval's theorem, one interpretation of it is that the total energy across all time is the total energy across all of its frequency components.

Also, from a mathematical standpoint what does this mean? Is this a manifestation of some property of integers?

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You may find these questions on MO interesting: mathoverflow.net/questions/14568/… and mathoverflow.net/questions/28826/… – Mike Spivey Nov 29 '10 at 22:10

The formula is not about the integers in the sense that it doesn't involve their multiplicative structure; rather, it's about how the integers sit inside the reals as a discrete subgroup. You can see this in the way the formula generalizes to $\mathbb{Z}^n$ sitting inside $\mathbb{R}^n$ or more generally in how it generalizes to the Selberg trace formula. The trace formula makes it clear that Poisson summation is a representation-theoretic fact; indeed one of its special cases is Frobenius reciprocity.

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If you want analogies from electrical engineering, one possible way to understand the consequences of the Poisson summation is the Nyquist sampling theorem. It says that periodic sampling of a signal is enough to capture all the signals whose frequency is less than or equal to half the sampling rate.

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But isn't it more of a consequence and manipulation of Poisson Summation formula? I am wondering if there is any fundamental law which manifests it in this way. Like Conservation of Energy manifests itself as Parseval's identity. – user17762 Nov 25 '10 at 11:17
Well, this is the only application of Poisson summation I know in electrical engineering. Conservation of energy is a physics statement. But even if the physical law turns out to be wrong(say in an alternate universe), Parseval's theorem would still be true. In general my feeling is that it is not completely correct to say that mathematical statements are manifestations of physical realities. For example Parseval's theorem is true in more generality as Plancherel theorem, en.wikipedia.org/wiki/Plancherel_theorem , which is not really a manifestation of any physical thing. – user1119 Nov 25 '10 at 11:30

Physical picture: The Poisson summation formula gives the Heatflow on the circle $\mathbb{R} / \mathbb{Z}$ when applied to $e^{-x^2 t}$ on the circle, that is a transformation law for the heat kernel

$$\sum\limits_{n} e^{-n^2 t} = \frac{1}{{2 \pi t}} \sum\limits_{n} e^{-n^2/4 t}.$$

Number theoretic picture: $$\theta ( t ) = \sum\limits_{n} e^{-n^2 i t}$$ can actually be defined for $\Im t >0$ and is a modular function. Using this gives nice proof of the functional equation for the Riemann zeta function.

Representation theoretic picture: The Poisson summation formula holds more generally for local compact abelian groups. The Selberg trace formula generalizes it to non commutative groups and cocompact discrete subgroups.

(Algebraic) Geometry: The Riemann Roch theorem for function fields follows from it, which can be seen as an index theorem.

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Also have a look at Tate's thesis, if you are interested in the modern way of analytic continuation of L functions. – plusepsilon.de Jun 8 '11 at 19:38

in QM the POISSON summation formula is interpreted as the density of energies

$\rho (E)= \sum_{n=0}^{\infty}\delta (x-n^{2} \pi ^{2})$ of the Hamiltonian operator

$H=p^{2} =-\hbar^{2} \frac{d^{2}}{dx^{2}}y(x)$ with boundary conditions $y(0)=0=y(1)$

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Poisson's sum formula is just an extension of Fourier series. The only difference is that Fourier series is for periodic function. But in Poisson's sum formula, f(n) are non periodic functions (or you can call them basis function).

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