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 $TopAb$ of locally compact Hausdorff abelian groups

$$0 \to A \to B \to C \to 0.$$

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

$$\int_B f(b)\; d\mu_B(b) = \int_C \int_A f(a + c)\; d\mu_A(a) d\mu_C(c)$$

for all continuous functions $f: B \to \mathbb{C}$ with compact support. (The inner integral on the right is a shorthand for $\int_A f(a + b)\; d\mu_A(a)$ for any $b \in B$ that maps to $c \in C$; this is well-defined since the integral is invariant under changes $b \mapsto b + a'$ within the same coset $c$.) We remark that given Haar measures $d\mu_A, d\mu_B$, there exists a Haar measure $d\mu_C$ 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 $d\mu_A, d\mu_B, d\mu_C$ determine the third.

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

Theorem

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

$$\int_A f(a)\; d\mu_A = \int_{\widehat{C}} \widehat{f}(\widehat{c})\; d\mu_{\widehat{C}}$$

where $\widehat{f}$ is the Fourier transform of $f$, as a function on $\widehat{B}$ defined by

$$\widehat{f}(\chi) = \int_B f(b)\overline{\chi(b)}d \mu_B(b).$$

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

$$\sum_{x \in L} f(x) = \frac1{\mu(B/L)} \sum_{y \in L^\perp} \widehat{f}(y)$$

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

The classical case is when $B$ is a Euclidean space $\mathbb{R}^n$. 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

• The Poisson summation formulation implies the functional equation for the Jacobi theta function, which in turn implies that of the Riemann zeta function, see at Riemann zeta function – functional identity.

Related entries

• functional equation

• A nonabelian version of the Poisson summation formula is the Selberg trace formula.

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:

• John Tate, Fourier analysis in number fields, and Hecke's zeta-functions, Princeton, May 1950, thesis; reproduced in Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) pp. 305–347, Academic Press 1967

  MR0217026

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

• Paul Garrett, Iwasawa-Tate on ζ-functions and L-functions, 2011 (pdf