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.
which decays at infinity with all derivatives (Schwartz function), the Poisson summation formula states that where
Then Eq.1 is a special case (P=1, x=0) of this generalization:[2][3] which is a Fourier series expansion with coefficients that are samples of the function
The Poisson summation formula can also be proved quite conceptually using the compatibility of Pontryagin duality with short exact sequences such as[7]
Eq.2 holds in the strong sense that both sides converge uniformly and absolutely to the same limit.
The Fourier series on the right-hand side of Eq.2 is then understood as a (conditionally convergent) limit of symmetric partial sums.
, but then it is necessary to interpret it in the sense that the right-hand side is the (possibly divergent) Fourier series of
[10] 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.
The Poisson summation formula similarly provides a connection between Fourier analysis on Euclidean spaces and on the tori of the corresponding dimensions.
The Poisson summation formula is also useful to bound the errors obtained when an integral is approximated by a (Riemann) sum.
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.
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.
Cohn & Elkies[14] 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.
The Poisson summation formula holds in Euclidean space of arbitrary dimension.
decay sufficiently fast at infinity, then one can "invert" the Fourier series back to their domain
itself are related by proper normalisation Note that the right hand side is independent of the choice of invariant measure
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 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, and James Arthur, have generalised the Poisson summation formula to the Fourier transform on non-commutative locally compact reductive algebraic groups
plays the role of the real number line in the classical version of Poisson summation, and
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 Poisson summation formula is the archetype for vast developments in harmonic analysis and number theory.
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 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.
The Poisson summation formula is a particular case of the convolution theorem on tempered distributions.