In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Poisson summation formula is an equation that relates the
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
coefficients of the
periodic summation
In signal processing, any periodic function s_P(t) with period ''P'' can be represented by a summation of an infinite number of instances of an aperiodic function s(t), that are offset by integer multiples of ''P''. This representation is called p ...
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
Baron Siméon Denis Poisson FRS FRSE (; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electri ...
and is sometimes called Poisson resummation.
Forms of the equation
Consider an aperiodic function
with
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
alternatively designated by
and
The basic Poisson summation formula is:
Also consider periodic functions, where parameters
and
are in the same units as
:
:
Then is a special case (P=1, x=0) of this generalization:
which is a
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
expansion with coefficients that are samples of function
Similarly:
also known as the important
Discrete-time Fourier transform
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values.
The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
.
The Poisson summation formula can also be proved quite conceptually using the compatibility of
Pontryagin duality
In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...
with
short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
s such as
:
[
]
Applicability
holds provided is a continuous integrable function
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...
which satisfies
:
for some and every [ Note that such is ]uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
, 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 In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
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 (conditionally convergent) limit of symmetric partial sums.
As shown above, holds under the much less restrictive assumption that is in , but then it is necessary to interpret it in the sense that the right-hand side is the (possibly divergent) 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 , case 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
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
, the Poisson summation formula provides a rigorous justification for the fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not a ...
of the heat equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for ...
with absorbing rectangular boundary by the method of images The method of images (or method of mirror images) is a mathematical tool for solving differential equations, in which the domain of the sought function is extended by the addition of its mirror image with respect to a symmetry hyperplane. As a resul ...
. Here the heat kernel
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectr ...
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
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
.
In electrodynamics
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
, the method is also used to accelerate the computation of periodic Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
s.
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
Bandlimiting is the limiting of a signal's frequency domain representation or spectral density to zero above a certain finite frequency.
A band-limited signal is one whose Fourier transform or spectral density has bounded support.
A bandli ...
, meaning that there is some cutoff frequency such that 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
The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals and discrete-time signals. It establishes a sufficient condition for a sample rate that per ...
.[
]
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
The Poisson summation formula is also useful to bound the errors obtained when an integral is approximated by a (Riemann) sum. Consider an approximation of as , where is the size of the bin. Then, according to this approximation coincides with . The error in the approximation can then be bounded as . This is particularly useful when the Fourier transform of is rapidly decaying if .
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
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalle ...
then [
]
Number theory
In number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
, Poisson summation can also be used to derive a variety of functional equations including the functional equation for the Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
.[ H. M. Edwards (1974). ''Riemann's Zeta Function''. Academic Press, pp. 209–11. .]
One important such use of Poisson summation concerns theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
s: 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
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory o ...
. By choosing and using the fact that one can conclude:
: 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
Cohn & Elkies[ proved an upper bound on the density of ]sphere packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three- dimensional Euclidean space. However, sphere pack ...
s using the Poisson summation formula, which subsequently led to a proof of optimal sphere packings in dimension 8 and 24.
Other
* Let for and for to get
* 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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
of arbitrary dimension. Let be the lattice in consisting of points with integer coordinates. 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 on lies in with
Moreover, for all in (Fourier transform on ) equals (Fourier transform on ).
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 above.
More generally, a version of the statement holds if Λ is replaced by a more general lattice in . The '']dual lattice
In the theory of lattices, the dual lattice is a construction analogous to that of a dual vector space. In certain respects, the geometry of the dual lattice of a lattice L is the reciprocal of the geometry of L , a perspective which underlie ...
'' Λ′ can be defined as a subset of the dual vector space or alternatively by Pontryagin duality
In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...
. 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 function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
s, and is a possible method in geometry of numbers Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in \mathbb R^n, and the study of these lattices provides fundamental informa ...
. In fact in more recent work on counting lattice points in regions it is routinely used − summing the indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
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
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
.
Selberg trace formula
Further generalization to locally compact abelian groups is required in number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
. In non-commutative harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...
, 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
Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarded ...
, 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.
Convolution theorem
The Poisson summation formula is a particular case of the convolution theorem
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g ...
on tempered distributions
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
. If one of the two factors is the Dirac comb
In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula
\operatorname_(t) \ := \sum_^ \delta(t - k T)
for some given period T. Here ''t'' is a real variable and th ...
, one obtains periodic summation
In signal processing, any periodic function s_P(t) with period ''P'' can be represented by a summation of an infinite number of instances of an aperiodic function s(t), that are offset by integer multiples of ''P''. This representation is called p ...
on one side and sampling on the other side of the equation. Applied to the Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
and its Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
, the function that is constantly 1, this yields the Dirac comb identity.
See also
*
* Post's inversion formula
In mathematics, the inverse Laplace transform of a function ''F''(''s'') is the piecewise-continuous and exponentially-restricted real function ''f''(''t'') which has the property:
:\mathcal\(s) = \mathcal\(s) = F(s),
where \mathcal denotes the ...
* Voronoi formula
In mathematics, a Voronoi formula is an equality involving Fourier coefficients of automorphic forms, with the coefficients twisted by additive characters on either side. It can be regarded as a Poisson summation formula for non-abelian groups. Th ...
* Discrete-time Fourier transform
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values.
The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
* Explicit formulae for L-functions In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by for the Riemann zeta function. Such explicit formulae have been applied a ...
References
Further reading
*
*
*
{{DEFAULTSORT:Poisson Summation Formula
Fourier analysis
Generalized functions
Lattice points
Theorems in analysis
Summability methods