Editing Poisson summation formula (section)

==Generalizations==
The Poisson summation formula holds in [[Euclidean space]] of arbitrary dimension. Let <math>\Lambda</math> be the [[lattice (group theory)|lattice]] in <math>\mathbb{R}^d</math> consisting of points with integer coordinates. For a function <math>s</math> in <math>L^1(\mathbb{R}^d)</math>, consider the series given by summing the translates of <math>s</math> by elements of <math>\Lambda</math>:

<math display="block">\mathbb{P}s(x) = \sum_{\nu\in\Lambda} s(x+\nu).</math>

'''Theorem''' For <math>s</math> in <math>L^1(\mathbb{R}^d)</math>, the above series converges pointwise almost everywhere, and defines a <math>\Lambda</math>-periodic function on <math>\mathbb{R}^d</math>, hence a function <math>\mathbb{P}s(\bar x)</math> on the torus <math>\mathbb{R}^d/\Lambda.</math> a.e. <math>\mathbb{P}s</math> lies in <math>L^1(\mathbb{R}^d/\Lambda)</math> with <math>\| \mathbb{P}s \|_{L_1(\mathbb{R}^d/\Lambda)} \le \| s \|_{L_1(\mathbb{R})}.</math><br>
Moreover, for all <math>\nu</math> in <math>\Lambda,</math> 
:<math>\mathbb{P}S(\nu) = \int_{\mathbb{R}^d/\Lambda}\mathbb{P}s(\bar x) e^{-i2\pi  \nu \cdot \bar x} d\bar x</math> 
(the Fourier transform of <math>\mathbb{P}s</math> on the torus <math>\mathbb{R}^d/\Lambda</math>) equals
:<math>S(\nu) = \int_{\mathbb{R}^d}s(x) e^{-i2\pi\nu \cdot x}\,dx</math>
(the Fourier transform of <math>s</math> on <math>\mathbb{R}^d</math>).

When <math>s</math> is in addition continuous, and both <math>s</math> and <math>S</math> decay sufficiently fast at infinity, then one can "invert" the Fourier series back to their domain <math>\mathbb{R}^d</math> and make a stronger statement. More precisely, if

<math display="block">|s(x)| + |S(x)| \le C (1+|x|)^{-d-\delta}</math>

for some ''C'', ''δ'' > 0, then<ref name="Stein"/>{{rp|at=VII §2}}

<math display="block">\sum_{\nu\in\Lambda} s(x+\nu) = \sum_{\nu\in\Lambda} S(\nu) e^{i 2\pi \nu\cdot x}, </math>

where both series converge absolutely and uniformly on Λ. When ''d'' = 1 and ''x'' = 0, this gives {{EquationNote|Eq.1}} above.

More generally, a version of the statement holds if Λ is replaced by a more general lattice in a finite dimensional vector space <math>V</math>. Choose a [[Haar Measure|translation invariant measure]] <math>m</math> on <math>V</math>. It is unique up to positive scalar. Again for a function <math>s \in L_1(V, m)</math> we define the periodisation

:<math> \mathbb{P}s(x) = \sum_{\nu \in \Lambda} s(x + \nu)</math>
as above. 

The ''[[dual lattice]]'' <math>\Lambda'</math> is defined as a subset of the [[dual vector space]] <math>V'</math> that evaluates to integers on the lattice <math>\Lambda</math> or alternatively, by [[Pontryagin duality]], as the characters of <math>V</math> that contain <math>\Lambda</math> in the kernel.
Then the statement is that for all <math>\nu \in \Lambda'</math> the Fourier transform <math>\mathbb{P}S</math> of the periodisation <math>\mathbb{P}s</math> as a function on <math>V/\Lambda</math> and the Fourier transform <math>S</math> of <math>s</math> on <math>V</math> itself are related by proper normalisation 
:<math>\begin{align}
\mathbb{P}S(\nu) &= \frac{1}{m(V/\Lambda)} \int_{V/\Lambda} \mathbb{P}s(\bar x) e^{-i2\pi\langle\nu, \bar x\rangle} m(d\bar x)\\
 &= \frac{1}{m(V/\Lambda)} \int_V s(x) e^{-i2\pi\langle\nu, x\rangle} m(dx) \\
 &= \frac{1}{m(V/\Lambda)} S(\nu)
\end{align}
</math>
Note that the right-hand side is independent of the choice of invariant measure <math>\mu</math>. If <math> s </math> and <math> S</math> are continuous and tend to zero faster than <math>1/r^{\dim(V) + \delta}</math> then 
:<math> \sum_{\lambda \in \Lambda} s(\lambda +x) = \sum_{\nu \in \Lambda'} \mathbb{P}S(\nu) e^{i2\pi\langle\nu, x\rangle} =  \frac{1}{m(V/\Lambda)} \sum_{\nu \in \Lambda'} S(\nu) e^{i2\pi\langle\nu, x\rangle}
</math>
In particular 
:<math> \sum_{\lambda \in \Lambda} s(\lambda)  =  \frac{1}{m(V/\Lambda)} \sum_{\nu \in \Lambda'} S(\nu) 
</math>

This is applied in the theory of [[theta function]]s 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 [[Sides of an equation|LHS]] of the summation formula is what is sought and the [[Sides of an equation|RHS]] something that can be attacked by [[mathematical analysis]].

===Selberg trace formula===
{{main article|Selberg trace formula}}
Further generalization to [[locally compact abelian group]]s 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 <math> G</math>  with a discrete subgroup <math> \Gamma</math>  such that <math> G/\Gamma</math>  has finite volume. For example, <math> G</math>  can be the real points of <math> SL_n</math>  and <math> \Gamma</math>  can be the integral points of <math> SL_n</math>. In this setting, <math> G</math>  plays the role of the real number line in the classical version of Poisson summation, and <math> \Gamma</math>  plays the role of the integers <math> n</math>  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 {{EquationNote|Eq.1}} becomes a sum over irreducible unitary representations of <math> G</math>, and is called "the spectral side," while the right-hand side becomes a sum over conjugacy classes of <math> \Gamma</math>, 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===
{{see also|Convolution theorem#Convolution theorem for tempered distributions}}
The Poisson summation formula is a particular case of the [[convolution theorem]] on [[Distribution (mathematics)#Tempered distribution|tempered distributions]]. If one of the two factors is the [[Dirac comb]], one obtains [[periodic summation]] on one side and [[Sampling (signal processing)|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#Dirac-comb identity|Dirac comb identity]].