Poisson summation formula


In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.

Forms of the equation

For appropriate functions the Poisson summation formula may be stated as:
With the substitution, and the Fourier transform property, , becomes:
With another definition, and the transform property becomes a periodic summation and its equivalent Fourier series:
Similarly, the periodic summation of a function's Fourier transform has this Fourier series equivalent:
where T represents the time interval at which a function is sampled, and is the rate of samples/sec.

Examples

These equations can be interpreted in the language of distributions for a function whose derivatives are all rapidly decreasing.
The Poisson Summation Formula arises as a particular case of the
Convolution Theorem on tempered distributions.
Using the Dirac comb distribution and its Fourier series:
In other words, the periodization of a Dirac delta,
resulting in a Dirac comb, corresponds to the discretization of its spectrum which is constantly one.
Hence, this again is a Dirac comb but with reciprocal increments.
readily follows:
Similarly:

Derivation

We can also prove that holds in the sense that if, then the right-hand side is the Fourier series of the left-hand side. This proof may be found in either or. It follows from the dominated convergence theorem that exists and is finite for almost every. And furthermore it follows that is integrable on the interval. The right-hand side of has the form of a Fourier series. So it is sufficient to show that the Fourier series coefficients of are. Proceeding from the definition of the Fourier coefficients we have:
The Poisson summation formula can also be proved quite conceptually using the compatibility of Pontryagin duality with short exact sequences such as

Applicability

holds provided is a continuous integrable function which satisfies
for some and every . Note that such is uniformly continuous, this together with the decay assumption on, show that the series defining converges uniformly to a continuous function. holds in the strong sense that both sides converge uniformly and absolutely to the same limit.
holds in a pointwise sense under the strictly weaker assumption that has bounded variation and
The Fourier series on the right-hand side of is then understood as a limit of symmetric partial sums.
As shown above, holds under the much less restrictive assumption that is in Lp space|, but then it is necessary to interpret it in the sense that the right-hand side is the Fourier series of . In this case, one may extend the region where equality holds by considering summability methods such as Cesàro summability. When interpreting convergence in this way holds under the less restrictive conditions that is integrable and 0 is a point of continuity of. However may fail to hold even when both and are integrable and continuous, and the sums converge absolutely.

Applications

Method of images

In partial differential equations, the Poisson summation formula provides a rigorous justification for the fundamental solution of the heat equation with absorbing rectangular boundary by the method of images. Here the heat kernel on is known, and that of a rectangle is determined by taking the periodization. The Poisson summation formula similarly provides a connection between Fourier analysis on Euclidean spaces and on the tori of the corresponding dimensions. In one dimension, the resulting solution is called a theta function.

Sampling

In the statistical study of time-series, if is a function of time, then looking only at its values at equally spaced points of time is called "sampling." In applications, typically the function is band-limited, meaning that there is some cutoff frequency such that the Fourier transform is zero for frequencies exceeding the cutoff: for. For band-limited functions, choosing the sampling rate guarantees that no information is lost: since can be reconstructed from these sampled values, then, by Fourier inversion, so can. This leads to the Nyquist–Shannon sampling theorem.

Ewald summation

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. This is the essential idea behind Ewald summation.

Lattice points in a sphere

The Poisson summation formula may be used to derive Landau's asymptotic formula for the number of lattice points in a large Euclidean sphere. It can also be used to show that if an integrable function, and both have compact support then .

Number theory

In number theory, Poisson summation can also be used to derive a variety of functional equations including the functional equation for the Riemann zeta function.
One important such use of Poisson summation concerns theta functions: periodic summations of Gaussians. Put, for a complex number in the upper half plane, and define the theta function:
The relation between and turns out to be important for number theory, since this kind of relation is one of the defining properties of a modular form. By choosing in the second version of the Poisson summation formula, and using the fact that, one gets immediately
by putting.
It follows from this that has a simple transformation property under and this can be used to prove Jacobi's formula for the number of different ways to express an integer as the sum of eight perfect squares.

Sphere packings

proved an upper bound on the density of sphere packings using the Poisson summation formula, which subsequently led to a proof of optimal sphere packings in dimension 8 and 24.

Generalizations

The Poisson summation formula holds in Euclidean space of arbitrary dimension. Let be the lattice in consisting of points with integer coordinates; is the character group, or Pontryagin dual, of. For a function in, consider the series given by summing the translates of by elements of :
Theorem For in, the above series converges pointwise almost everywhere, and thus defines a periodic function Pƒ on. Pƒ lies in with ||Pƒ||1 ≤ ||ƒ||1. Moreover, for all in, Pƒ̂ equals .
When is in addition continuous, and both and decay sufficiently fast at infinity, then one can "invert" the domain back to and make a stronger statement. More precisely, if
for some C, δ > 0, then
where both series converge absolutely and uniformly on Λ. When d = 1 and x = 0, this gives the formula given in the first section above.
More generally, a version of the statement holds if Λ is replaced by a more general lattice in. The dual lattice Λ′ can be defined as a subset of the dual vector space or alternatively by Pontryagin duality. 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.

Selberg trace formula

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.
A series of mathematicians applying harmonic analysis to number theory, most notably Martin Eichler, Atle Selberg, Robert Langlands, and James Arthur, have generalised the Poisson summation formula to the Fourier transform on non-commutative locally compact reductive algebraic groups with a discrete subgroup such that has finite volume. For example, can be the real points of and can be the integral points of. In this setting, plays the role of the real number line in the classical version of Poisson summation, and plays the role of the integers that appear in the sum. The generalised version of Poisson summation is called the Selberg Trace Formula, and has played a role in proving many cases of Artin's conjecture and in Wiles's proof of Fermat's Last Theorem. The left-hand side of becomes a sum over irreducible unitary representations of, and is called "the spectral side," while the right-hand side becomes a sum over conjugacy classes of, and is called "the geometric side."
The Poisson summation formula is the archetype for vast developments in harmonic analysis and number theory.