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\section*{Poisson summation formula}

\hypertarget{idea}{}\subsection*{{Idea}}\label{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).

\hypertarget{statement}{}\subsection*{{Statement}}\label{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 space|locally compact]] [[Hausdorff space|Hausdorff]] [[topological abelian group|abelian groups]]

\begin{displaymath}
0 \to A \to B \to C \to 0.
\end{displaymath}
where $A, B, C$ are equipped with Haar measures $d\mu_A, d\mu_B, d\mu_C$ that make the following equation true:

\begin{displaymath}
\int_B f(b)\; d\mu_B(b) = \int_C \int_A f(a + c)\; d\mu_A(a) d\mu_C(c)
\end{displaymath}
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.

\begin{theorem}
\label{}\hypertarget{}{}
Let $\widehat{C}$ denote the [[Pontryagin duality|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

\begin{displaymath}
\int_A f(a)\; d\mu_A = \int_{\widehat{C}} \widehat{f}(\widehat{c})\; d\mu_{\widehat{C}}
\end{displaymath}
where $\widehat{f}$ is the Fourier dual of $f$, as a function on $\widehat{B}$.

\end{theorem}
In the special case of a [[lattice (discrete subgroup)|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

\begin{displaymath}
\sum_{x \in L} f(x) = \frac1{\mu(B/L)} \sum_{y \in L^\perp} \widehat{f}(y)
\end{displaymath}
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 \hyperlink{Tate1950}{Tate's thesis}, where a [[global field]] is viewed as a lattice inside its [[ring of adeles]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item 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 \emph{\href{Riemann+zeta+function#functional_equation}{Riemann zeta function -- functional identity}}.

\end{itemize}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[functional equation]]


\item A nonabelian version of the Poisson summation formula is the [[Selberg trace formula]].



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Reviews include

\begin{itemize}%
\item theorem 4.1 in \emph{Analytic theory of modular forms} \href{http://www.math.harvard.edu/~jbland/ma259x_notes.pdf}{pdf}


\item E. Kowalski, prop. 2.2.1 in \emph{Automorphic forms, L-functions and number theory (March 12--16) Three Introductory lectures} (\href{http://www.math.ethz.ch/~kowalski/lectures.pdf}{pdf})



\end{itemize}
An application to [[zeta functions]] via harmonic analysis on [[adele rings]] originates in Tate's thesis:

\begin{itemize}%
\item [[John Tate]], \emph{[[Fourier analysis in number fields, and Hecke's zeta-functions]]}, Princeton, May 1950, thesis; reproduced in \emph{Algebraic Number Theory} (Proc. Instructional Conf., Brighton, 1965) pp. 305--347, Academic Press 1967 \href{http://www.ams.org/mathscinet-getitem?mr=0217026}{MR0217026}

\end{itemize}
A textbook account is

\begin{itemize}%
\item [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of \emph{Automorphic representations and L-functions for the general linear group}, Cambridge Studies in Advanced Mathematics 129, 2011 (\href{https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf}{pdf})

\end{itemize}
and brief review in

\begin{itemize}%
\item [[Paul Garrett]], \emph{Iwasawa-Tate on $\zeta$-functions and L-functions}, 2011 (\href{http://www-users.math.umn.edu/~garrett/m/mfms/notes_c/Iwasawa-Tate.pdf}{pdf} [[!redirects Poisson formula]]

\end{itemize}


\end{document}