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.

which decays at infinity with all derivatives (Schwartz function), the simplest version of the Poisson summation formula states that is the Fourier transform The summation formula can be restated in many equivalent ways, but a simple one is the following. (L1 for L1 space is a unimodular lattice or otherwise.)

Periodization of a function

which decays at infinity with all derivatives (Schwartz function), and its Fourier transform

Then is a special case (P=1, x=0) of this generalization:

which is a Fourier series Conversely, follows from by applying the known behavior of the Fourier transform under translations (see the Fourier transform properties time scaling and shifting).

Similarly:

also known as the important Discrete-time Fourier transform.

Derivations

We prove that, is a Schwartz function, this establishes equality in of the previous section.

is the Fourier transform.)

Proceeding from the definition of the Fourier coefficients we have:

where the interchange of summation with integration is justified by dominated convergence. With a change of variables), this becomes the following, completing the proof of :

This proves , a similar proof shows the corresponding version of .

has an absolutely convergent Fourier series, then is a Schwartz function. Similarly, is a Schwartz function.

Distributional formulation

These equations can be interpreted in the language of distributionss whose derivatives are all rapidly decreasing (see Schwartz function). 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:

Or:

The Poisson summation formula can also be proved quite conceptually using the compatibility of Pontryagin duality with short exact sequences such as

Applicability

is a continuous integrable function is uniformly continuous 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 has bounded variation and

The Fourier series on the right-hand side of is then understood as a (conditionally convergent) limit of symmetric partial sums.

As shown above, is in L^1(\mathbb{R}) 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 . However, 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 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.

In electrodynamics, the method is also used to accelerate the computation of periodic Green's functions.

Sampling

is band-limited 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. (A broad function in real space becomes a narrow function in Fourier space and vice versa.) This is the essential idea behind Ewald summation.

Approximations of integrals

is the size of the bin. Then, according to .

Lattice points inside a sphere

both have compact support

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 a complex number in the upper half plane, and define the theta function:

turns out to be important for number theory, since this kind of relation is one of the defining properties of a modular form one can conclude:

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

Cohn & Elkies 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.

Other

• It can be used to prove the functional equation for the theta function.

• Poisson's summation formula appears in Ramanujan's notebooks and can be used to prove some of his formulas, in particular it can be used to prove one of the formulas in Ramanujan's first letter to Hardy.

• It can be used to calculate the quadratic Gauss sum.

Generalizations

The Poisson summation formula holds in Euclidean space be the lattice:

Theorem-periodic function

) equals

).

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 above.

. Choose a translation invariant measure we define the periodisation

as above.

The dual lattice is defined as a subset of the dual vector space or alternatively, by Pontryagin duality itself are related by proper normalisation

then

In particular

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 generalization 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 plays the role of the real number 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 , and is called "the geometric side."

The Poisson summation formula is the archetype for vast developments in harmonic analysis and number theory.

Semiclassical trace formula

The Selberg trace formula was later generalized to more general smooth manifolds (without any algebraic structure) by Gutzwiller, Balian-Bloch, Chazarain, Colin de Verdière, Duistermaat-Guillemin, Uribe, Guillemin-Melrose, Zelditch and others. The "wave trace" or "semiclassical trace" formula relates geometric and spectral properties of the underlying topological space. The spectral side is the trace of a unitary group of operators (e.g., the Schrödinger or wave propagator) which encodes the spectrum of a differential operator and the geometric side is a sum of distributions which are supported at the lengths of periodic orbits of a corresponding Hamiltonian system. The Hamiltonian is given by the principal symbol of the differential operator which generates the unitary group. For the Laplacian, the "wave trace" has singular support contained in the set of lengths of periodic geodesics; this is called the Poisson relation.

Convolution theorem

The Poisson summation formula is a particular case of the convolution theorem on tempered distributions. If one of the two factors is the Dirac comb, one obtains periodic summation on one side and sampling on the other side of the equation. Applied to the Dirac delta function and its Fourier transform, the function that is constantly 1, this yields the Dirac comb identity.

See also

• Post's inversion formula

• Voronoi formula

• Discrete-time Fourier transform

• Explicit formulae for L-functions

References

Further reading