Poisson summation formula

The Poisson summation formula is an equation relating the Fourier series coefficients of the periodic extension of a function to values in the function's continuous Fourier transform. In other words, the periodic extension of a function in one domain is completely defined by discrete samples of the original function in the other domain. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.

Definition

For appropriate functions, ƒ, the Poisson summation formula may be stated as:

$\underbrace {\sum _{n=-\infty }^{\infty }f(t+nT)} _{\varphi _{T}(t)}={\frac {1}{T}}\sum _{k=-\infty }^{\infty }{\hat {f}}\left({\frac {k}{T}}\right)\ e^{2\pi i{\frac {k}{T}}t},$ (Eq.1)

where ${\hat {f}}$ is the Fourier transform of $f,\,$  $\varphi _{T}$ is a periodic extension of $f,\,$ and $T$ is the period of $\varphi _{T}.$

For t=0  Eq.1 reduces to:

$\sum _{n=-\infty }^{\infty }f(nT)={\frac {1}{T}}\sum _{k=-\infty }^{\infty }{\hat {f}}\left({\frac {k}{T}}\right),$ (Eq.2)

which holds under the less restrictive conditions that 0 is a point of continuity of φ_T(t). This may fail to be the case even when both $f$ and ${\hat {f}}$ are continuous and the sums converge absolutely (Katznelson 1976).

Derivation of the Poisson summation formula

The right-hand side of Eq.1 has the form of a Fourier series. So it is sufficient to show that the Fourier series coefficients of $\varphi _{T}(t)$ are ${\frac {1}{T}}\ {\hat {f}}\left({\frac {k}{T}}\right)$, as follows:

${\begin{aligned}{\hat {\varphi }}_{T}[k]\ &{\stackrel {\mathrm {def} }{=}}\ {\frac {1}{T}}\int _{0}^{T}\varphi _{T}(t)\cdot e^{-2\pi i{\frac {k}{T}}t}\,dt\\&={\frac {1}{T}}\int _{0}^{T}\left(\sum _{n=-\infty }^{\infty }f(t+nT)\right)\cdot e^{-2\pi i{\frac {k}{T}}t}\,dt\\&={\frac {1}{T}}\sum _{n=-\infty }^{\infty }\int _{0}^{T}f(t+nT)\cdot e^{-2\pi i{\frac {k}{T}}t}\,dt\end{aligned}}$

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

${\begin{aligned}{\hat {\varphi }}_{T}[k]&={\frac {1}{T}}\sum _{n=-\infty }^{\infty }\int _{nT}^{nT+T}f(\tau )\ e^{-2\pi i{\frac {k}{T}}(\tau -nT)}d\tau \\&={\frac {1}{T}}\sum _{n=-\infty }^{\infty }\int _{nT}^{nT+T}f(\tau )\ e^{-2\pi i{\frac {k}{T}}\tau }\underbrace {\ e^{i2\pi kn}} _{=1\forall k,n}d\tau \\&={\frac {1}{T}}\sum _{n=-\infty }^{\infty }\int _{nT}^{nT+T}f(\tau )\ e^{-2\pi i{\frac {k}{T}}\tau }d\tau \\&={\frac {1}{T}}\int _{-\infty }^{\infty }f(\tau )\ e^{-2\pi i{\frac {k}{T}}\tau }d\tau \quad {\stackrel {\mathrm {def} }{=}}\quad {\frac {1}{T}}\ {\hat {f}}\left({\frac {k}{T}}\right).\end{aligned}}$

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

• J.J. Benedetto; G. Zimmermann: Sampling multipliers and the Poisson summation formula. J. Fourier Ana. App. 3(1997)5, Preprint online
• J.R. Higgins: Five short stories about the cardinal series. Bull. AMS 12(1985)1, Online at Project Open Euclid
• Grafakos, Loukas (2004), Classical and Modern Fourier Analysis, Pearson Education, Inc., pp. 253–257, ISBN 0-13-035399-X.
• Katznelson, Yitzhak (1976), An introduction to harmonic analysis (Second corrected ed.), New York: Dover Publications, Inc, ISBN 0-486-63331-4
• Pinsky, M. (2002), Introduction to Fourier Analysis and Wavelets., Brooks Cole.
• Zygmund, Antoni (1968), Trigonometric series (2nd ed.), Cambridge University Press (published 1988), ISBN 978-0521358859.