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\title{General Mathematics and Computational Science II}
\author{Exercise 19}
\date{May 3, 2007}
\maketitle

\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 \,.
  \tag{*}
\end{equation*}

\item The previous question suggests a way to differentiate a function
$u \colon [0, 2\pi] \to \mathbb R$ numerically: First, sample $u$ on
an equidistant grid.  Second, compute the Fourier coefficients of its
trigonometric interpolant $v$ as described in class.  Third, use (*)
on the Fourier coefficients $\hat v_k$.  Finally, take the inverse DFT
to obtain the derivative sampled on the grid.

The problem is that any standard software library DFT assumes that
$j=0,\dots,N-1$ and $k=0,\dots,N-1$ and will store the arrys in
precisely this order.  Trigonometric interpolation, on the other hand,
requires that $k=-N/2,\dots,N/2-1$.  With which factor, then, must the
$i$th element, with respect to the order imposed by the software, of
the vector $\hat {\boldsymbol v}$ be muliplied so that differentiation
is computed correctly?

\item Solve the difference equation
\begin{gather*}
  t_n = 2 \, t_{n-1} + 2^{n+1} \,, \\
  t_1 = 0 \,.
\end{gather*}




\end{enumerate}
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