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\title{General Mathematics and Computational Science II}
\author{Exercise 21}
\date{May 10, 2007}
\maketitle

\begin{enumerate}

\item Let $G$ be a finite abelian group, written additively, and let
$\hat G$ denote its dual group.  Show that for every nonzero $a \in
G$,
\[
  \sum_{\chi \in \hat G} \chi(a) = 0 \,.
\]
\emph{Hint:} Mimic the proof of the corresponding statement on $G$ as
given in class.

\item Prove the Plancharel identity
\[
  \langle \hat f, \hat g \rangle = n \, \langle f,g \rangle \,,
\]
with the inner products
\begin{gather*}
  \langle f, g \rangle 
  = \frac1n \sum_{a \in G} \overline{f(a)} \, g(a) \,, \\
  \langle \hat f, \hat g \rangle
  = \frac1n \sum_{\chi \in \hat G} 
    \overline{\hat f(\chi)} \, \hat g(\chi) \,,
\intertext{and Fourier transform}
  \hat f(\chi) = \sum_{a \in G} \chi(a) \, f(a) \,.
\end{gather*}
  


\item Let $f_A \in \mathbb C^G$ denote the characteristic function of
a set $A \subset G$, and $\hat f_A \in \mathbb C^{\hat G}$ its Fourier
transform.  Set
\[
  \Phi(A) = \max \{ \lvert \hat f_A(\chi) \rvert \colon
                    \chi \in \hat G, \chi \neq \chi_0 \} \,.
\]

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


\end{enumerate}
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