nLab
Poisson summation formula

Idea

A certain identity for certain Fourier transforms, equating a sum or integral of a function over a domain (e.g., a lattice) with a corresponding sum or integral of its Fourier dual over a dual domain (e.g., the dual lattice).

Statement

The Poisson summation formula is basic to harmonic analysis over general locally compact Hausdorff abelian groups.

Consider an exact sequence in TopAbTopAb of locally compact Hausdorff abelian groups

0ABC0.0 \to A \to B \to C \to 0.

where A,B,CA, B, C are equipped with Haar measures dμ A,dμ B,dμ Cd\mu_A, d\mu_B, d\mu_C that make the following equation true:

Bf(b)dμ B(b)= C Af(a+c)dμ A(a)dμ C(c)\int_B f(b)\; d\mu_B(b) = \int_C \int_A f(a + c)\; d\mu_A(a) d\mu_C(c)

for all continuous functions f:Bf: B \to \mathbb{C} with compact support. (The inner integral on the right is a shorthand for Af(a+b)dμ A(a)\int_A f(a + b)\; d\mu_A(a) for any bBb \in B that maps to cCc \in C; this is well-defined since the integral is invariant under changes bb+ab \mapsto b + a' within the same coset cc.) We remark that given Haar measures dμ A,dμ Bd\mu_A, d\mu_B, there exists a Haar measure dμ Cd\mu_C making this Fubini-type equation true. Then, since Haar measures form a torsor over the group of positive reals with multiplication, it follows that any two of dμ A,dμ B,dμ Cd\mu_A, d\mu_B, d\mu_C determine the third.

In this notation, the “Poisson summation formula” is the equation asserted by the following result.

Theorem

Let C^\widehat{C} denote the Pontryagin dual of CC, and dμ C^d\mu_{\widehat{C}} the dual Haar measure. For any Schwartz-Bruhat function f:Bf: B \to \mathbb{C}, we have

Af(a)dμ A= C^f^(c^)dμ C^\int_A f(a)\; d\mu_A = \int_{\widehat{C}} \widehat{f}(\widehat{c})\; d\mu_{\widehat{C}}

where f^\widehat{f} is the Fourier dual of ff, as a function on B^\widehat{B}.

In the special case of a lattice LL inside BB, the dual space L =B/L^L^\perp = \widehat{B/L} is a lattice inside B^\widehat{B}, and the integrals are over discrete spaces, i.e. integration is just summation and we have

xLf(x)=1μ(B/L) yL f^(y)\sum_{x \in L} f(x) = \frac1{\mu(B/L)} \sum_{y \in L^\perp} \widehat{f}(y)

where μ\mu is the Haar measure on B/LB/L (as above). Often the measure on BB is chosen so that μ(B/L)=1\mu(B/L) = 1.

The classical case is when BB is a Euclidean space n\mathbb{R}^n. But another case of a lattice inside locally compact abelian groups occurs in the context of Tate’s thesis, where a global field is viewed as a lattice inside its ring of adeles.

Examples

References

Reviews include

  • theorem 4.1 in Analytic theory of modular forms pdf

  • E. Kowalski, prop. 2.2.1 in Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures (pdf)

An application to zeta functions via harmonic analysis on adele rings originates in Tate’s thesis:

A textbook account is

  • Dorian Goldfeld, Joseph Hundley, chapter 2 of Automorphic representations and L-functions for the general linear group, Cambridge Studies in Advanced Mathematics 129, 2011 (pdf)

and brief review in

Last revised on February 15, 2018 at 12:08:44. See the history of this page for a list of all contributions to it.