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.
Consider an aperiodic function formula_1 with Fourier transform formula_2 alternatively designated by formula_3 and formula_4 
The basic Poisson summation formula is:
Also consider periodic functions, where parameters formula_5 and formula_6 are in the same units as formula_7:
Then is a special case (P=1, x=0) of this generalization:
which is a Fourier series expansion with coefficients that are samples of function formula_9 Similarly:
also known as the important Discrete-time Fourier transform.
_{1}\,d\tau
\ =\ \frac{1}{P} \int_{-\infty}^{\infty} s(\tau) \ e^{-i 2\pi \frac{k}{P} \tau} d\tau \triangleq \frac{1}{P}\cdot S\left(\frac{k}{P}\right)
\end{align}&lt;/math&gt; 
Distributional formulation
These equations can be interpreted in the language of distributions for a function formula_10 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 formula_12
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.
For the case formula_13 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.
 holds provided formula_1 is a continuous integrable function which satisfies
for some formula_20 and every formula_21 Note that such formula_1 is uniformly continuous, this together with the decay assumption on formula_10, show that the series defining formula_24 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 formula_10 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 formula_1 is in formula_28, but then it is necessary to interpret it in the sense that the right-hand side is the (possibly divergent) Fourier series of formula_29 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 formula_30 holds under the less restrictive conditions that formula_1 is integrable and 0 is a point of continuity of formula_32. However may fail to hold even when both formula_10 and formula_34 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 formula_35 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.
In the statistical study of time-series, if formula_10 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 formula_10 is band-limited, meaning that there is some cutoff frequency formula_38 such that formula_39 is zero for frequencies exceeding the cutoff: formula_40 for formula_41 For band-limited functions, choosing the sampling rate formula_42 guarantees that no information is lost: since formula_34 can be reconstructed from these sampled values. Then, by Fourier inversion, so can formula_44 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.
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 formula_45 as formula_46, where formula_47 is the size of the bin. Then, according to this approximation coincides with formula_48. The error in the approximation can then be bounded as formula_49. This is particularly useful when the Fourier transform of formula_50 is rapidly decaying if formula_51.
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, formula_10 and formula_34 both have compact support then formula_54
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 formula_55, for formula_56 a complex number in the upper half plane, and define the theta function:
The relation between formula_58 and formula_59 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 formula_60 and using the fact that formula_61 one can conclude:
It follows from this that formula_64 has a simple transformation property under formula_65 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 &amp; 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.
Generalizations.
The Poisson summation formula holds in Euclidean space of arbitrary dimension. Let formula_71 be the lattice in formula_72 consisting of points with integer coordinates. For a function formula_10 in formula_74, consider the series given by summing the translates of formula_10 by elements of formula_71:
Theorem For formula_10 in formula_74, the above series converges pointwise almost everywhere, and thus defines a periodic function formula_80 on formula_81  formula_80 lies in formula_83 with formula_84&lt;br&gt;
Moreover, for all formula_85 in formula_86  formula_87 (Fourier transform on formula_71) equals formula_89 (Fourier transform on formula_72).
When formula_10 is in addition continuous, and both formula_10 and formula_34 decay sufficiently fast at infinity, then one can "invert" the domain back to formula_72 and make a stronger statement. More precisely, if
for some "C", "δ" &gt; 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 formula_72. 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 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 formula_98 with a discrete subgroup formula_99 such that formula_100 has finite volume. For example, formula_98 can be the real points of formula_102 and formula_99 can be the integral points of formula_102. In this setting, formula_98 plays the role of the real number line in the classical version of Poisson summation, and formula_99 plays the role of the integers formula_107 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 formula_98, and is called "the spectral side," while the right-hand side becomes a sum over conjugacy classes of formula_99, 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 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.