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\title{General Mathematics and CPS II}
\author{Exercise 19}
\date{May 2, 2012}
\maketitle


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\begin{enumerate}


\item Handout, Exercise 3.

\item Suppose a smooth periodic function $u \colon [0,2\pi) \to
\mathbb C$ is sampled at the $N$ equidistant points $x_j=jh$ with $h =
2 \pi/N$ and $j = 0, \dots, N-1$.  We set $u_j = u(x_j)$, take the
discrete Fourier transform
\[
  \tilde u_k = \sum_{j=0}^{N-1} \e^{-\i k x_j} \, u_j \,,
\]
and define the reconstruction of $u$ via
\[
  v(x) = \sum_{k=-N/2}^{N/2-1} \e^{\i k x} \, \tilde u_k \,.
\]
Show that
\[
  \hat v_k = 
  \begin{cases}
    \tilde u_k & \text{for } k \in -N/2, \dots, N/2-1 \,, \\
    0 & \text{otherwise} \,.
  \end{cases}
\]


\end{enumerate}
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