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\title{General Mathematics and Computational Science II}
\author{Exercise 18}
\date{April 26, 2007}
\maketitle

\noindent
Recall that the discrete Fourier transform of an $N$-tuple of numbers
$u_1, \dots, u_N$ is defined
\[
  \hat u_k = \frac1N \sum_{j=1}^N \e^{-\i k x_j} \, u_j
\]
for $k=1,\dots, N$, where $x_j = jh$ with $h=2\pi/N$.

\begin{enumerate}

\item Prove \emph{Parseval's identity}
\[
  \sum_{k=1}^N \lvert \hat u_k \rvert^2
  = \frac1N \sum_{j=1}^N \lvert u_j \rvert^2 \,.
\]

\item Show that $\hat u_{k+N} = \hat u_k$.  Vice versa, setting
$u_{j+N}=u_j$, show that
\[
  \hat u_k = \frac1N \sum_{j=m+1}^{m+N} \e^{-\i k x_j} \, u_j
\]
for every integer $m$.

\item Let
\[
  w_j = \sum_{l=1}^N u_l \, v_{j-l}
\]
with the understanding that the three $N$-tuples are periodically
extended beyond their basic range of definition on which the index
varies from $1$ to $N$; cf.\ Question~2.

Show that
\[
  \hat w_k = \hat u_k \, \hat v_k \,.
\]

\end{enumerate}
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