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& \textbf{Real Analysis} & \hspace{9mm} \\
April 2006 \hspace{10mm} & \textbf{Ph.D. Qualifying Exam} & R. Nagisetty \& Denis White \\
&\textbf{Answer any six questions.}& \\\end{tabular}
\begin{enumerate}
\item
For $n=1,2,3 \ldots$, let $f_{n}(x) = (1/n)\arctan(n^2 x^2)$ for $x$
in $\mathbb{R}$. Show that $f_{n}$ converges uniformly on the
entire real line and moreover that the sequence $f_{n}'$ of
derivatives converges pointwise on all of $\mathbb{R}$. Show also
that $f_{n}'$ does not converge uniformly on any interval containing
the origin.
\item
Prove or disprove the following. Suppose $(a_{n})_{n \ge 1}$ is
a sequence of real numbers such that $\lim_{n \to \infty} a_{n} =0$
and such that the partial sums
\[
S_{N} = \sum_{n=1}^{N} a_{n}
\]
are bounded for every positive integer $N$. Then $\sum_{n=1}^{\infty} a_{n}$
converges.
\item
\begin{enumerate}
\item State the Stone Weierstrass Theorem.
\item Let $X$ and $Y$ be compact Hausdorff spaces and let $C(X)$
denote the space of continuous functions defined on $X$. For
$f \in C(X)$ and $g \in C(Y) $ define the function $f \otimes g$
on $X \times Y$ by $f \otimes g(x,y)= f(x)g(y)$. Show that every continuous
function defined on $X \times Y$ can be approximated uniformly by
a finite sum $\sum_{i=1}^{N} f_{i} \otimes g_{i}$ where $f_{i} \in C(X)$
and $g_{i} \in C(Y)$.
\end{enumerate}
\item
Suppose that $f \in L^{1}(\mathbb{R})$ and $g$, defined by $g(x) = xf(x)$
is also in $L^{1}(\mathbb{R})$. Show that $\hat{f}$, defined by
\[
\hat{f}(\xi) = \int_{-\infty}^{\infty} e^{-ix \xi}f(x) \, dx,
\]
for $ \xi \in \mathbb{R}$ is continuously differentiable and
its derivative $\hat{f}'(\xi)$ is bounded.
\item
\begin{enumerate}
\item
State the definition of equicontinuous for a set of continuous functions.
\item
State the Arzela-Ascoli Theorem.
\item
Suppose that $K$ is a continuous real valued function
defined on a square $[a,b] \times [a,b]$ and define $T$ on
the space $C[a,b]$ of continuous functions defined on $[a,b]$
by
\[
Tf(x) = \int_{a}^{b} K(x,y) f(y) \, dy
\]
Show that the image under $T$ of a bounded set in $C[a,b]$
has compact closure in $C[a,b]$.
\end{enumerate}
\item
Suppose that $f \in L^{1}(\mathbb{R})$. Show that, for every
$\epsilon>0$ there is $\delta>0$ so that, if a Borel set $E$ has
Lebesgue measure at most $\delta$ then
\[
|\int_{E}f(x) \, dx| < \epsilon
\]
\item
Let $f(x)=\ln(1+x).$ Derive the MacLaurin series for $f(x)$ and show
that it converges back to $f(x)$ in the interval $(-1,1)$.
\item
Define a convex function on the real line and show that it is
differentiable at all but a countable set of points.
\item
Let $s_{n+1}=\sqrt{2+\sqrt{s_n}}$ be a recurrence relation with
$s_1>0$. Show that the sequence $\{s_n\}$ converges.[Show that it is
monotone and bounded.]
\end{enumerate}
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