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\begin{document}

{\bf Name:} \underline{\mbox{\hspace*{7cm}}}
\hfill{\bf Instructor:} Jeffrey Diller

\begin{center}
{\bf Math 366: Honors Real Analysis II}
\\
{\bf Spring Semester 2004}
\\
{\bf Final Exam}
\\
{\bf Wednesday, May 5}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} examination contains 5 problems.  Counting
the front cover and blank pages, the exam consists of 6 sheets
of paper.  
\\[1.5ex]
\hline
\end{tabular}

\begin{center} {\bf Scores}

\begin{tabular}{|c|c|c|}\hline
Question & Possible\quad &\ Actual\quad\rule[-.3cm]{0cm}{0.9cm} \\ \hline\hline
  1      &  35  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  2      &  40  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  3      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  4      &  10  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  5      &  10  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  6      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  7      &  15  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  8      &  10  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
Total  & 140\mbox{\  \ } & \rule[-.3cm]{0cm}{0.9cm} \\ \hline
\end{tabular}
\end{center}

\vspace*{.5cm}
\begin{center}     {\bf GOOD LUCK}\end{center}
\newpage

\begin{problems}

\item Do all of the following five (7 points each).  
\begin{subproblem}
\item Let $E\subset\R$ be a set.  Define the \emph{outer measure}
of $E$.
\vspace*{2in}

\item State the \emph{dominated convergence theorem}.
\vspace*{2in}

\item Define \emph{measurable subset of $\R$}.
\vspace*{1in}

\item Define the linear space $L^1$ (i.e. $L^1(\R)$) together with its norm.
\newpage

\item State Fejer's Theorem---define any relevant Fourier series notation.
\vspace*{4in}
\end{subproblem}

\item On the next page give examples of four of the following five
(10 points each).  To define a function, it is enough to draw its graph.
\begin{subproblem}
\item 
A sequence of continuous functions $f_n:\R\to\R$  
that
\begin{itemize}
\item $f_n$ converges pointwise but not uniformly to a continuous function
$f:\R\to\R$.
\item $\int f = \lim\int f_n$.
\end{itemize}
\item 
Same as previous, except that $\int f \neq \lim\int f_n$.
\item 
A continuous non-negative integrable function $f:\R\to\R$ such that 
$\limsup_{x\to\infty} f(x) = \infty$.
\item 
A sequence of functions $f_n:\R\to\R$ such that
\begin{itemize}
\item $\lim_{n\to\infty} f_n = 0$ in $L^1$;
\item $\lim_{n\to\infty} f_n(x)$ does not exist at any point in $\R$.
\end{itemize}
It suffices here to define the first several $f_n$'s---enough to see
the pattern.
\item 
A sequence of sets $E_n\subset\R$ such that $E_{n+1}\subset E_n$ and
$mE_n = \infty$ for every $n\in\N$, but $m(\bigcap_{n\in\N} E_n) = 0$.
\end{subproblem}

\newpage
\item Compute the fourier coefficients of the function 
$f:[-\frac12,\frac12]\to\R$ 
given by $f(x) = |x|$ (15 points).

\newpage
\item Show that a set $E\subset\R$ with $m^*E = 0$ is measureable.
(10 points) \vspace*{3in}

\item State and prove Fatou's lemma (10 points). 

\newpage
\item Let $f:[-\frac12,\frac12]\to\R$ be an $L^2$ function.  Prove that
among all trig polynomials of the form $\sum_{n=-N}^N c_n e_n(x)$,
the fourier approximation $S_N f$ is the best approximation as
measured by the $L^2$ metric (15 points).

\newpage
\item Let $f:\R\to\R$ be a non-negative integrable function and 
$F:\R\to\R$ be given by
$$
F(x) = \int_{(-\infty,x)} f.
$$
Show that $F$ is continuous (15 points).

\newpage
\item For ten points extra credit, do one of the following two:
\begin{subproblem}
\item Let $f:\R\to\R$ be a $1$-periodic, $C^2$ function.  Show that
the fourier approximations $S_Nf$ converge uniformly to $f$ as $N\to\infty$.
(Hint: what is the relationship between fourier coefficients of $f$ and
$f''$.)
\item Let $f_n:[0,1]\to\R$ be measurable functions converging pointwise
to a function $f:[0,1]\to\R$.  Show that for any $\epsilon > 0$, 
there exists $A\subset [0,1]$ such that $mA > 1-\epsilon$ and $f_n\to f$
uniformly on $A$.
\end{subproblem}
\end{problems}
\end{document}
\end
