# How does one use the Poisson summation formula?

While reading the answer to another Mathoverflow question, which mentioned the Poisson summation formula, I felt a question of my own coming on. This is something I've wanted to know for a long time. In fact, I've even asked people, who have probably given me perfectly good answers, but somehow their answers have never stuck in my brain. The question is simple: the Poisson summation formula is incredibly useful to many people, but why is that? When you first see it, it looks like a piece of magic, but then suddenly you start spotting that people keep saying "By Poisson summation" and expecting you to fill in the details. In that respect, it's a bit like the phrase "By compactness," but the important difference for me is that I can fill in the details of compactness arguments.

What I would like to know is this. What is the "trigger" that makes people think, "Ah, Poisson summation should be useful here"? And is there some very simple example of how it is applied, with the property that once you understand that example, you basically understand how to apply it in general? (Perhaps two or three examples are needed -- that would obviously be OK too.) And can one give a general description of the circumstances where it is useful? (Anyone familiar with the Tricki will see that I am basically asking for a Tricki article on the formula. But I don't mind something incomplete or less polished.)

For reference, here is a related (but different) question about the Poisson summation formula: Truth of the Poisson summation formula

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The existing answers list some important situations where Poisson Summation plays a role, the application to proving the functional equation of $\theta$ and hence of $\zeta$ being my personal favourite. My best answer to Tim's question as he actually asked it might be: why not have it in mind to try using it whenever you have a discrete sum that you are having trouble estimating, especially if you fancy your chances of understanding the Fourier transform of the summands. You'll end up with a different sum and it might be a lot easier to understand, and you might even be able to approximate your first sum by an integral (the term $\hat{f}(0)$ in the Poisson summation formula).

To explain a little more with an example, there's a whole theory concerned with the estimation of exponential sums $\sum_{n \leq N} e^{2\pi i \phi(n)}$. There are two processes called A and B that can be used to turn a sum like this into something you might be better positioned to understand. Process A is basically Weyl/van der Corput differencing (Cauchy-Schwarz) and process B is essentially Poisson summation. It's not a very straightforward task to put together a theory of how these processes interact, and how they may best be combined to estimate your sum, and in fact this is in general something of an art. The 10 lectures book by Montgomery contains a nice exposition, and there's a whole LMS lecture note volume by Graham and Kolesnik if you want more details.

I want to share a perhaps slightly obscure paper of Roberts and Sargos (Three-dimensional exponential sums with monomials, Journal fur die reine und angewandte Mathematik (Crelle) 591), in which they use Poisson Summation in the form of Process B mentioned above arbitrarily many times to establish the following rather simple-to-state result: the number of quadruples $x_1,x_2,x_3,x_4$ in $[X, 2X)$ with

$$|1/x_1 + 1/x_2 - 1/x_3 - 1/x_4| \leq 1/X^3$$

is $X^{2 + o(1)}$. In other words, the quantities $1/a + 1/b$ tend to avoid one another to pretty much the same extent as random numbers of the same size. Very very roughly speaking (I don't really understand the argument in depth) the proof involves looking at exponential sums $\sum_x e^{2\pi i m/x}$, and it is these that are transformed repeatedly using Poisson summation followed by other modifications (it being reasonably pointless to try and apply Poisson sum twice in succession).

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That answer is indeed much more the kind of thing I was looking for: not so much how it has been used in the past (interesting though that is) but how it might be used in the future. –  gowers Jun 20 '10 at 21:12
There's an awful lot more that could be said. For instance, the PSF is a very baby instance of a trace formula, and more exotic trace formulae such as Selberg's are terribly important for relating "spectral" information and "geometric" information. Obviously, however, I'm not the right person to write about that... –  Ben Green Jun 20 '10 at 21:23

Perhaps surprisingly, I don't use Poisson summation per se all that often, but I do repeatedly use the more general principle that Poisson summation exemplifies, namely that the Fourier transform intertwines restriction and projection. Restricting a function $f: G \to {\bf C}$ on a group G to a subgroup H corresponds to projecting out the Fourier transform $\hat f: \hat G \to {\bf C}$ along the orthogonal complement $H^\perp$ (all the characters in $\hat G$ that annihilate H). Conversely, averaging out f along H corresponds to restricting $\hat f$ to $H^\perp$. Plugging in $G = {\bf R}$ and $H = {\bf Z}$ more or less gives the classical Poisson summation formula.

The same principle also applies to approximate groups, such as balls: localising a function in physical space to a ball of scale r corresponds to averaging the Fourier transform at scale 1/r, and conversely; this already explains the uncertainty principle to a large extent, and also is the starting point for Littlewood-Paley theory. Or, localising the zeroes of the Riemann zeta function to a strip of height T corresponds to understanding the distribution of the (logarithm of the) primes at scales 1/T and above; understanding the evolution of the wave equation up to time T controls the eigenvalues of the associated Laplacian at scales 1/T and above; and so on and so forth. So, in general, I know that fine scale behaviour of physical space is tied to coarse scale behaviour of frequency space and vice versa; and whenever I need to formalise this sort of intuition, I have to reach for something like a Poisson summation formula (though, as I said above, I rarely use that formula directly, but usually some variant that is jury-rigged from convolutions and cutoff operators).

[To interpret the classical Poisson summation formula as a variant of the uncertainty principle, one has to use an "adelic" topology and view the integers in physical space as "small", and the integers in frequency space as "low frequencies". I personally find this perspective helpful, but this may be because I come from a real-variable harmonic analysis background.]

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That "more general principle that Poisson summation exemplifies" is Poisson summation when G is finite. For a function f:G-->C, define its H-average and H-cutoff to be functions on G given by Avg_H(f)(g) = (1/#H)\sum_{h \in H} f(gh) and Cut_H(f)(g) = [f(g) when g is in H and 0 when g is not in H]. Then by a direct calculation, the Fourier transform of the function Avg_H(f) on G is the function Cut_{H^\perp}(f^) on G^, and if you take the Fourier transform of both sides to get an identity of functions on G you get the Poisson summation formula (Here f^(chi) = \sum_{g in G} f(g)chi*(g).) –  KConrad Jun 26 '10 at 21:18

In number theory, the Poisson summation formula is used in proving functional equations of theta functions (assuring that they are modular forms) and then via this proving functional equations for zeta- and L-functions. What is happening in both cases is that there is some fundamental duality (often self-duality) that is being exploited.

In coding theory there is also the MacWilliams identities which follow from a discrete version of Poisson summation. In some sense they are equivalent to certain restricted classes of functional equations for theta functions. Again duality and self-duality are behind these symmetries.

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Probably different answers in harmonic analysis and number theory.

Place a Dirac measure at each of the integers and you get a distribution that is close to being self-dual under the Fourier transform on the real line. As a Gaussian is. This starts ringing bells as potentially useful? But in an algebraic way, I suppose.

In number theory, we respect Poisson summation as the real reason the functional equation of the Riemann zeta-function is true. (Weil unified this remark with the first one). It has been said that Riemann thought of that proof as an exercise: it was easily within reach of the sort of thing the French school of applied maths had been doing for a generation. (Who, after all, was Poisson, and why did he bother?) Apparently separately, Siegel proposed Poisson summation as a technique in geometry of numbers. This is the version for a lattice in Euclidean space, now (already needed in Hecke's work on L-functions). I think the point there is that geometry of numbers already had other techniques, so that the basic idea of Fourier transform applied to the characteristic function of a set, and then Poisson summation to get a handle on the lattice points as a sum, wasn't needed earlier. But this idea has by now got fed into lattice-point counting as a routine step (I think - I'm not an exponential sum expert).

My ideas on this are probably not broad enough, but the underlying Gaussian-PS hybrid pretty much defines what a theta-function is in real-analysis terms. And what theta-functions are otherwise goes back to Weyl's thematics on the canonical commutation relations as Lie algebra. Can be read in numerous ways.

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This is really a comment that could be added on to existing answers, but I can't add comments. As a particularly simple example that follows Ben's answer, as well as an example of SandeepJ's observation about changing the 'size' of the sum, consider counting solutions to some linear equation L = 0 in a subset A of a finite abelian group G. Let's say $L = x + y + z$. To count solutions to $L(x,y,z) = 0$ with $x,y,z \in A$ one could look at the sum $\sum_{(x, y, z) \in H} F(x, y, z)$, where $F(x, y, z) = 1_A(x)1_A(y)1_A(z)$ and $H \leq G^3$ is the subgroup of solutions $(x,y,z) \in G^3$ such that $L(x,y,z) = 0$. Up to some normalization factors, Poisson summation then says something like $$\sum_{\mathbf{x} \in H} F(\mathbf{x}) = \sum_{\chi \in H^\perp} \widehat{F}(\chi) = \sum_{\gamma \in \widehat{G}} \widehat{1_A}(\gamma)^3,$$ since $H^\perp \cong \widehat{G}$. So the 'two-dimensional' sum on the left has become a 'one-dimensional' sum on the right.

This is of course the basis of a lot of arguments in additive combinatorics, and as Ben said one might fruitfully consider the term $\widehat{F}(0)$ in the sum on the right-hand side above.

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The Poisson summation formula for finite Abelian groups equates the sum over the subgroup H of G to the sum over its dual(H)=H#. This is used when when one sum is much larger than the other.

For a torus, it is a relation between the eigenvalues of the Laplacian and the lengths of closed geodesics.

This is explained in Audrey Terras's books (vol 1 and 2) on Harmonic Analysis.

She also applies the formula to the Ising model in Statistical Mechanics, Boolean switching functions and Macwilliams identities (the last has already been pointed out earlier by Robin Chapman above)

Stein and Shakarchi in their book Fourier Analysis apply the formula to relate Poisson kernels of the disc and the upper half plane, and the heat kernels of R/Z and R

These applications are in addition to those listed on wikipedia

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