Poisson Summation Formula - Generalizations

Generalizations

The Poisson summation formula holds in Euclidean space of arbitrary dimension. Let Λ be the lattice in Rd consisting of points with integer coordinates; Λ is the character group, or Pontryagin dual, of Rd. For a function ƒ in L1(Rd), consider the series given by summing the translates of ƒ by elements of Λ:

Theorem For ƒ in L1(Rd), the above series converges pointwise almost everywhere, and thus defines a periodic function Pƒ on Λ. Pƒ lies in L1(Λ) with ||Pƒ||₁ ≤ ||ƒ||₁. Moreover, for all ν in Λ, Pƒ̂(ν) (Fourier transform on Λ) equals ƒ̂(ν) (Fourier transform on Rd).

When ƒ is in addition continuous, and both ƒ and ƒ^ decay sufficiently fast at infinity, then one can "invert" the domain back to Rd and make a stronger statement. More precisely, if

for some C, δ > 0, then

(Stein & Weiss 1971, VII §2)

where both series converge absolutely and uniformly on Λ. When d = 1 and x = 0, this gives the formula given in the first section above.

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. 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.

Further generalisation 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.