Poisson summation formula 

The Poisson summation formula is an equation relating the coefficients of the Fourier series of the periodic extension of a function in terms of the values of the function's continuous Fourier transform. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.

Contents

Definition

The Poisson summation formula may be stated as: (Pinsky 2002) (Zygmund 1968)

\sum_{n=-\infty}^\infty f(nT)=\frac{1}{T}\sum_{n=-\infty}^\infty \hat f\left(\frac{n}{T}\right),

where

\hat{f}(\nu)\ \stackrel{\mathrm{def}}{=}\int_{-\infty}^{\infty} f(x)\ e^{-2\pi i\nu x}\, dx.

There is a second version of the Poisson summation formula that is equivalent for nice enough functions, and which relies on the notion of a periodic extension.

The Periodic extension of a function is a new function formed by shifting the original function by multiples of some period, T, and adding all the copies together. The periodic extension of a function ƒ(t) can be written:

\varphi_T(t)\ \stackrel{\mathrm{def}}{=}\ \sum_{n=-\infty}^{\infty} f(t + nT).

Assuming f is integrable and that \sum_{k=-\infty}^{\infty}\left|\hat f\left(\frac{k}{T}\right)\right|<\infty,  it can be shown below that: (Grafakos 2004)

\varphi_T(t) = \frac{1}{T} \sum_{k=-\infty}^{\infty} \hat f\left(\frac{k}{T}\right)\ e^{2\pi i \frac{k}{T} t}. 

 

 ( Eq.1)

 

For the special case t=0  Eq.1  reduces to the version of the Poisson summation formula given above. The first version holds under the less restrictive conditions that 0 is a point of continuity of φT(t). This may fail to be the case even when both f and \hat{f} are continuous and the sums converge absolutely (Katznelson 1976).

Intuitive derivation of the Poisson summation formula

The function φT(t) is periodic, with period T. It can therefore be expanded into a Fourier series,whose coefficients are given by:

\begin{align} \hat\varphi_T[k] \ &\stackrel{\mathrm{def}}{=}\ \frac{1}{T}\int_0^{T} \varphi_T(t)\cdot e^{-2\pi i \frac{k}{T} t}\, dt\\ &= \frac{1}{T}\int_0^{T} \left(\sum_{n=-\infty}^{\infty} f(t + nT)\right)\cdot e^{-2\pi i\frac{k}{T} t}\, dt\\ &= \frac{1}{T} \sum_{n=-\infty}^{\infty} \int_0^{T} f(t + nT)\cdot e^{-2\pi i\frac{k}{T} t}\, dt \end{align}

With a change of variables (\tau \ \stackrel{\mathrm{def}}{=}\ t + nT) this becomes:

\begin{align} \hat\varphi_T[k] & = \frac{1}{T} \sum_{n=-\infty}^{\infty} \int_{nT}^{nT + T} f(\tau) \ e^{-2\pi i\frac{k}{T} (\tau - nT)} d\tau \\ & = \frac{1}{T} \sum_{n=-\infty}^{\infty} \int_{nT}^{nT + T} f(\tau) \ e^{-2\pi i\frac{k}{T} \tau } \underbrace{\ e^{i 2\pi k n}}_{=1 \forall k, n} d\tau \\ & = \frac{1}{T} \sum_{n=-\infty}^{\infty} \int_{nT}^{nT + T} f(\tau) \ e^{-2\pi i\frac{k}{T} \tau} d\tau \\ & = \frac{1}{T} \int_{-\infty}^{\infty} f(\tau) \ e^{-2\pi i\frac{k}{T} \tau} d\tau\quad \stackrel{\mathrm{def}}{=}\quad \frac{1}{T}\ \hat f\left(\frac{k}{T}\right). \end{align}

Substitution of these coefficients into the Fourier series produces  Eq.1 .

Applications of the Poisson summation formula

The Poisson Summation formula may be used to give a proof of the Shannon Sampling theorem (Pinsky 2002). It also provides a connection between Fourier analysis on the circle and the real line. (Grafakos 2004).

Computationally, the Poisson summation formula is useful since a slowly converging summation in real space is guaranteed to be converted into a quickly converging equivalent summation in Fourier space. (A broad function in real space becomes a narrow function in Fourier space and vice versa.) This is the essential idea behind Ewald summation.

Convergence conditions

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

\Delta(t) \ = \ \sum_{n=-\infty}^{\infty} \delta(t - n)

summed over all integers n, we have that Δ is a distribution (a so-called Dirac comb), because applied to any test function we get a bi-infinite sum that has very small 'tails'. Then one may interpret the summation formula as saying that \Delta(t) \ 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, for constant a ≠ 0 it would follow that

\Delta(at) \ is the Fourier transform of \Delta(t/a) \ .

Therefore we can always find a 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 centered at each integer, calculate using the known Fourier transform of a Gaussian, and then let the width of all the Gaussians become small.

Generalizations

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.

Literature