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\title{General Mathematics and CPS II}
\author{Exercise 21}
\date{April 29, 2015}
\maketitle


There is no class Friday, May 1.  This homework set is due Wednesday,
Mai 6 in class.
\begin{enumerate}

\item Let $u \colon [0, 2\pi] \to \mathbb R$ be defined via the
Fourier series
\[
  u(x) = \sum_{k \in \mathbb Z} \hat u_k \, \e^{\i k x} \,.
\]
Show that its derivative $w(x) = u'(x)$, assuming it exists, has 
Fourier coefficients
\begin{equation*}
  \hat w_k = \i k \, \hat u_k \,.
\end{equation*}

\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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