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\begin{document}

\title{Real Analysis}
\author{Homework 10}
\date{due November 19, 2008}
\maketitle

\begin{enumerate}

\item  
\begin{enumerate}
\item Show that for $f \in L^1(\mathbb{R}^n)$,
\[
  \lim_{y\to 0}
  \int_{\mathbb{R}^n} |f(x) - f(x-y)| \, dx =0 \,.
\]

\emph{Hint:} Use mollifiers.

\item Suppose that, moreover, $g \in L^1(\mathbb{R}^n) \cap
  L^\infty(\mathbb{R}^n)$.  Show that $f*g \in C(\mathbb{R}^n) \cap
  L^1(\mathbb{R}^n)$.

\end{enumerate}

\item Prove \emph{Parseval's identity}
\[
  \int_{\R^n} u(x) \, \overline v(x) \, \d x
  = \int_{\R^n} \hat u(\xi) \, \overline {\hat v} (\xi) \, \d \xi \,.
\]

\emph{Note:} No need to prove from scratch, you may derive this from
the \emph{Plancharel identity} 
\[
  \lVert u \rVert_2 = \lVert \hat u \rVert_2
\]
by a polarization argument.

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

\emph{Hint:} Note that
\[
  (f( \, \cdot \, - y))^{\hat{}}(\xi) = \e^{- \i y \cdot \xi} \, \hat
  f(\xi) \,.
\]

\item Show that the definition of the Fourier transform on $L^2$ does
  not depend on the choice of the approximating sequence.

\end{enumerate}

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