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).
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
where $A, B, C$ are equipped with Haar measures $d\mu_A, d\mu_B, d\mu_C$ that make the following equation true:
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.
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
where $\widehat{f}$ is the Fourier dual of $f$, as a function on $\widehat{B}$.
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
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.
A nonabelian version of the Poisson summation formula is the Selberg trace formula.
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
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.