Poisson summation formula

Poisson summation formula

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Poisson summation formula is an equation that relates the Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...

coefficients of the periodic summation of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

to values of the function's continuous Fourier transform
Continuous Fourier transform
The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum. For instance, the transform of a musical chord made up of pure notes is a mathematical representation of the amplitudes of the individual notes that make...

. 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
Siméon Denis Poisson
Siméon Denis Poisson , was a French mathematician, geometer, and physicist. He however, was the final leading opponent of the wave theory of light as a member of the elite l'Académie française, but was proven wrong by Augustin-Jean Fresnel.-Biography:...

and is sometimes called Poisson resummation.

Forms of the equation

For appropriate functions ƒ, the Poisson summation formula may be stated as:
where   is the Fourier transform of ;  that is

With the substitution,   and the Fourier transform property,    (for P > 0),   becomes :
With another definition,    and the transform property    becomes a periodic summation (with period P) 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 s(t) is sampled, and 1/T is the rate of samples/sec.

Applicability

A simple demonstration of the feasibility of is as follows:

The quantity in parentheses is zero for all values of x except those of the form x=k, where k is any integer. At those values the summation diverges at a rate that is independent of k. The summation can effectively be replaced by an infinite sequence of equal-strength Dirac delta functions (called Dirac comb
Dirac comb
In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions...

), and we continue accordingly.

The normalizing factor, A, is readily shown to be 1 by considering the symmetrical case

Conditions that ensure is applicable are that ƒ is a continuous integrable function
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

which satisfies
for some C, δ > 0 and every x . Note that such ƒ is uniformly continuous; this together with the decay assumption on ƒ, show that the series defining SP converges uniformly to a continuous function. holds in the strong sense that both sides converge uniformly and absolutely to the same limit .

also holds in a pointwise
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.-Definition:...

sense under the strictly weaker assumption that ƒ 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.

holds under the much less restrictive assumption that ƒ is in L1(R)
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

, but then it is necessary to interpret it in the sense that the right-hand side is the (possibly divergent) Fourier series of SP(t) . In this case, one may extend the region where equality holds by considering summability methods such as Cesàro summability
Cesàro summation
In mathematical analysis, Cesàro summation is an alternative means of assigning a sum to an infinite series. If the series converges in the usual sense to a sum A, then the series is also Cesàro summable and has Cesàro sum A...

. When interpreting convergence in this way holds under the less restrictive conditions that ƒ is integrable and 0 is a point of continuity of SP(t). However may fail to hold even when both and are integrable and continuous, and the sums converge absolutely .

holds in the sense that if ƒ ∈ L1(R), then the right-hand side is the (possibly divergent) Fourier series of the left-hand side. This proof may be found in either or . It follows from the dominated convergence theorem
Dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions...

that SP(t) exists and is finite for almost every t and furthermore it follows that SP is integrable on the interval [0,P].

Applications

In partial differential equations, the Poisson summation formula provides a rigorous justification for the fundamental solution
Fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function...

of the heat equation
Heat equation
The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...

with absorbing rectangular boundary by the method of images
Method of images
See also Method of image charges for applications in electrostatics and magnetostaticsMethod of 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, with...

. Here the heat kernel
Heat kernel
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a particular domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some...

on R2 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 signal processing, the Poisson summation formula leads to the Discrete-time Fourier transform
Discrete-time Fourier transform
In mathematics, the discrete-time Fourier transform is one of the specific forms of Fourier analysis. As such, it transforms one function into another, which is called the frequency domain representation, or simply the "DTFT", of the original function . But the DTFT requires an input function...

and the Nyquist–Shannon sampling theorem
Nyquist–Shannon sampling theorem
The Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon, is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal into a numeric sequence...

.

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
Ewald summation
Ewald summation, named after Paul Peter Ewald, is a method for computing the interaction energies of periodic systems , particularly electrostatic energies. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an...

.

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   .

Poisson summation can also be used to derive a variety of functional equations including the functional equation for the Riemann zeta function.

Distributional formulation

can be interpreted in the language of distribution
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...

s . Let δ(t) be the Dirac delta function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

. Define

summed over all integers n. It can be shown that ΔP is a tempered distribution
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...

, known as the Dirac comb
Dirac comb
In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions...

: intuitively, this follows since applied to any Schwartz function one gets a bi-infinite series whose tails decay rapidly. One may then interpret the summation formula as the formula:
That is, up to an overall normalization, ΔP is its own Fourier transform. Indeed, if ƒ is a Schwartz function, then applying to ƒ gives precisely ; alternatively, taking a convolution with ƒ gives precisely . If ƒ is a function or distribution that is sufficiently regular that such a convolution can be defined in the distribution sense, then holds in the sense of distributions.

Generalizations

A version of the Poisson summation formula holds in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

with only slight modifications. Let Λ be the lattice in Rd consisting of points with integer coordinates. Then for suitable functions ƒ, one has

As in the case of one variable, this holds in the sense of Fourier series if ƒ is assumed to be integrable, and pointwise if ƒ satisfies the decay condition

for some C, δ > 0.

More generally, a version of the statement holds if Λ is replaced by a more general lattice in Rd. The dual lattice Λ′ can be defined as a subset of the dual vector space or alternatively by Pontryagin duality
Pontryagin duality
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.-Introduction:...

. 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
Geometry of numbers
In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by ....

. 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
Sides of an equation
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. Each is solely a name for a term as part of an expression; and they are in practice interchangeable, since equality is symmetric...

of the summation formula is what is sought and the RHS
Sides of an equation
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. Each is solely a name for a term as part of an expression; and they are in practice interchangeable, since equality is symmetric...

something that can be attacked by mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

.

Further generalisation to locally compact abelian groups is required in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

. In non-commutative harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...

, the idea is taken even further in the Selberg trace formula
Selberg trace formula
In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of G on the space L2 of square-integrable functions, where G is a Lie group and Γ a cofinite discrete group...

, but takes on a much deeper character.