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\begin{document}

\title{Real Analysis}
\author{Homework 11}
\date{due November 26, 2008}
\maketitle

\begin{enumerate}

\item Compute the Fourier transform on $\R$ of $\chi_{[-1,1]}(x)$.

\item (From Folland.) Let $f(x) = x$ on $[-\pi,\pi)$, interpreted as a
  function on $\T$.
  \begin{enumerate}
  \item Compute the Fourier series of $f$.
  \item Show that 
\[
  \sum_{k=1}^\infty \frac1{k^2} = \frac{\pi^2}6 \,.
\]
\emph{Hint:} Parseval identity.
  \end{enumerate}

\item Prove the \emph{Riemann--Lebesgue lemma} for Fourier series,
  i.e.\ for $f \in L^1(\T^n)$, $\hat f_k \to 0$ as $\lvert k \rvert
  \to \infty$.

\item (From Folland.) Let
\[
  D_m(x) = \frac1{2\pi} \sum_{\lvert k \rvert \leq m} \e^{\i kx}
\]
be the $m$th Dirichlet kernel.  Show that
\[
  \lVert D_m \rVert_{L^1(\T)} \to \infty
  \qquad \text{as} \qquad
  m \to \infty \,.
\]


\end{enumerate}

\end{document}
