Poisson summation formula

From Academic Kids

In mathematics, the Poisson summation formula (PSF) is a relation holding between a sum of a function F over all integers, and a corresponding sum for the Fourier transform G in its incarnation <math>G(\omega)=\int_{\mathbb R}F(x)e^{-i2\pi\cdot\omega x}\,dx<math>. If the normalization of the Fourier transform is correctly adjusted, it takes the form

Σ F(n) = Σ G(n).

It is also called (mainly by physicists) Poisson resummation. Its discoverer is Simeon Poisson.

Note: For the more frequently used form of the Fourier transform

<math>\hat F(\omega):=\frac1{\sqrt{2\pi}}G\left(\frac{\omega}{2\pi}\right)=\frac1{\sqrt{2\pi}}\int_{\mathbb R}F(x)e^{-i\omega x}\,dx<math>

the PSF reads

<math>\sum_{n\in\mathbb Z}F(n)={\sqrt{2\pi}}\sum_{k\in\mathbb Z}\hat F(2\pi\,k)<math>.

Some conditions restricting F must naturally be applied to have convergence here. A useful way to get around stating those precisely is to use the language of distributions. Let δ(x) be the Dirac delta function. Then if we write

Δ(x) = Σ δ(x − n)

summed over all integers n, we have that Δ is a distribution in good standing, because applied to any test function we get a bi-infinite sum that has very small 'tails'. Then a neat way to restate the summation formula is to say that

Δ is its own Fourier transform.

Again this depends on precise normalization in the transform; but it conveys good information about the variance of the formula. For example it is easy to see that for constant a ≠ 0 it would follow that

Δ(ax) is the Fourier transform of Δ(x/a).

Therefore we can always find some spacing λZ of the integers, such that placing a delta-function at each of those points is its own transform, and each normalization will have a corresponding valid formula. It also suggests a method of proof that is intuitive: put instead a Gaussian centred at each integer, calculate using the known Fourier transform of a Gaussian, and then let the width of all the Gaussians become small.

There is a version in n dimensions, that is easy to formulate. Given a lattice Λ in Rn, there is a dual lattice Λ′ (defined by vector space or Pontryagin duality, as one wishes). Then the statement is that the sum of delta-functions at each point of Λ, and at each point of Λ′, are again Fourier transforms as distributions, subject to correct normalization.

This is applied in the theory of theta functions, and is a possible method in geometry of numbers. In fact in more recent work on counting lattice points in regions it is routinely used − summing the indicator function of a region D over lattice points is exactly the question, so that the LHS of the summation formula is what is sought and the RHS something that can be attacked by mathematical analysis.

Further generalisation to locally compact abelian groups is required in number theory. In non-commutative harmonic analysis, the idea is taken even further in the Selberg trace formula, but takes on a much deeper character.


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