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.
Consider an aperiodic function
s(x)
\hats(f)
l{F}\{s\}(f).
The basic Poisson summation formula is:
Also consider periodic functions, where parameters
T>0
P>0
x
Then is a special case (P=1, x=0) of this generalization:which is a Fourier series expansion with coefficients that are samples of the function
S(f).
also known as the important Discrete-time Fourier transform.
The Poisson summation formula can also be proved quite conceptually using the compatibility of Pontryagin duality with short exact sequences such as
holds provided
s(x)
C>0,\delta>0
x.
s(x)
s
s | |
P |
holds in a pointwise sense under the strictly weaker assumption that
s
As shown above, holds under the much less restrictive assumption that
s(x)
L1(R)
s | |
P |
(x).
x=0,
s(x)
s | |
P |
(x)
s
S
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
R2
In electrodynamics, the method is also used to accelerate the computation of periodic Green's functions.
In the statistical study of time-series, if
s
s
fo
S(f)
S(f)=0
|f|>fo.
\tfrac{1}{T}>2fo
S
s.
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.[1] (A broad function in real space becomes a narrow function in Fourier space and vice versa.) This is the essential idea behind Ewald summation.
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
\delta\ll1
s(x)
1/\delta\gg1
The Poisson summation formula may be used to derive Landau's asymptotic formula for the number of lattice points inside a large Euclidean sphere. It can also be used to show that if an integrable function,
s
S
s=0.
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.[2]
One important such use of Poisson summation concerns theta functions: periodic summations of Gaussians . Put
q=ei\pi
\tau
The relation between
\theta(-1/\tau)
\theta(\tau)
s(x)=
-\pix2 | |
e |
S(f)=
-\pif2 | |
e |
,
by putting
{1/λ}=\sqrt{\tau/i}.
It follows from this that
\theta8
\tau\mapsto{-1/\tau}
Cohn & 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.
s(x)=e-ax
0\leqx
s(x)=0
x<0
The Poisson summation formula holds in Euclidean space of arbitrary dimension. Let
Λ
Rd
s
L1(Rd)
s
Λ
Theorem For
s
L1(Rd)
Ps
Λ.
Ps
L1
\|Ps\|1\le\|s\|1.
\nu
Λ,
PS(\nu)
Λ
S(\nu)
Rd
When
s
s
S
Rd
for some C, δ > 0, thenwhere 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
Rd
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.
See main article: article and 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
G
\Gamma
G/\Gamma
G
SLn
\Gamma
SLn
G
\Gamma
n
G
\Gamma
The Poisson summation formula is the archetype for vast developments in harmonic analysis and number theory.
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.