\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,graphicx}

\newcommand{\R}{\mathbb{R}}
\newcommand{\ghathat}{{\widehat{\widehat{G\,}}\!}}

\begin{document}

\title{General Mathematics and CPS II}
\author{Exercise 23}
\date{May 9, 2014}
\maketitle

\begin{enumerate}

\item Let $G$ be a finite abelian group of order $N$, written
additively, let $\widehat G$ denote its dual group, and $\ghathat$ the
bi-dual group (i.e., the set of characters on $\widehat G$).

\begin{enumerate}
\item Show that for every fixed $a \in G$ the map
\[
  \chi \mapsto \chi(a)
\]
defines an element of $\ghathat$.

\item Show that the construction in part (a) in fact defines a group
isomorphism between $G$ and $\ghathat$.
\end{enumerate}


\item Let $G$ be a finite abelian group of order $N$, written
additively, and let $\widehat G$ denote its dual group.  Show that for
every nonzero $a \in G$,
\[
  \sum_{\chi \in \widehat G} \chi(a) = 0 \,.
\]
Then conclude that
\[
  \sum_{\chi \in \widehat G} \overline{\chi(a)} \chi(b) 
  =
  \begin{cases}
    N & \text{if } a=b \,, \\
    0 & \text{otherwise} \,.
  \end{cases}
\]

\emph{Hint:} Mimic the proof of the corresponding orthogonality
relation for characters as given in class.

\item Let $f_A \in \mathbb C^G$ denote the characteristic function of
a set $A \subset G$, i.e.,
\[
  f_A(a) =
  \begin{cases}
  1 & \text{if } a \in A \,, \\
  0 & \text{if } a \notin A \,,
  \end{cases}
\]
and define its \emph{Fourier transform} $\widehat f_A \in \mathbb C^{\widehat
G}$ via
\[
  \widehat f_A(\chi) = \sum_{a \in G} \chi(a) \, f_A(a) \,.
\]
Set
\[
  \Phi(A) = \max \{ \lvert \widehat f_A(\chi) \rvert \colon
                    \chi \in \widehat G, \chi \neq \chi_0 \} \,.
\]
(Recall that $\chi_0$ denotes the principal character where
$\chi_0(a)=1$ for all $a \in G$.)

Show that $\Phi(A) = \Phi(G \setminus A)$.

\end{enumerate}
\end{document}
