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\centerline{\bf Ph.D. Qualifying Exam}
\bigskip
\centerline{Fall 2001}
\bigskip
\noindent {\bf Instructions:}
\begin{itemize}
\item[1.] If you think that a problem is incorrectly stated ask the proctor. If her/his explanation is not to
your satisfaction, interpret the problem as you see fit, but not so that the answer is trivial.
\item[2.] From each part solve 3 of the 5 five problems.
\item[3.] If you solve more than three problems from each part, indicate the problems that you wish to have graded.
\end{itemize}
\bigskip
\centerline{\bf Part A}
\medskip\noindent
1. Suppose that $\ds \sum_{i=1}^{\infty} s_i$ is a series of positive terms with the property that $\ds
\sum_{i=1}^n s_i < \log(n)$. Show that $\ds \sum_{i=1}^\infty\frac{s_i}{i}$ converges.
\bigskip\noindent
2. Consider a convergent sequence $\{a_n\}_{n=0}^\infty$ with $\lim_{n\to\infty} a_n= a$. Let
\[
\alpha_n=\frac{a_0+\cdots +a_n}{n},
\]
and show that $\lim_{n\to\infty} \alpha_n = a$.
\bigskip\noindent
3. Denote the Banach space of absolutely convergent series of real numbers by $\ell^1$. An absolutely convergent
series $\sum_{i=1}^{\infty} a_i$ is said to converge at a rate determined by a positive convergent series
$\sum_{i=1}^\infty b_i$ if there is a real number $K$ such that for all $i$, $|a_i|1$
and $q$ arbitrary, or $r=1$ and $q>1$. Let $S$ be the set of all absolutely convergent series that do not converge
at a rate determined by positive series of this form. Show that $S$ is a dense subset of $\ell^1$.
\bigskip\noindent
4. Suppose that $f(x)$ is a uniform limit of step functions defined on the closed interval $[a,b]$. Prove that at any
point of $[a,b]$ the right and left limits of $f(x)$ exist.
\bigskip\noindent
5. (a) For any partition $\mathcal{P}=\{x_0,\ldots,x_n\}$ of an interval $[a,b]$ and any $f(x)$ defined on $[a,b]$
define the variation of $f(x)$ relative to $\mathcal{P}$ by
\[
V(f,\mathcal{P})=\sum_{i=0}^{n-1}|f(x_{i+1})-f(x_i)|
\]
and define the total variation of $f(x)$ by $V(f)=\sup_{\mathcal{P}}V(f,\mathcal{P})$. The space of functions that
have finite total variation is called the space of functions of bounded variation and is denoted by $BV([a,b])$.
Show that if $f\in BV([a,b])$ then $||f||=|f(0)|+V(f)$ defines a norm on $BV([a,b])$. (Correction: $||f||=|f(a)|+V(f)$.)
(b) Show that if a subset of $BV([a,b])$ is open in the sup norm, then it is open in the norm defined in (a).
\bigskip
\centerline{\bf Part B}
\medskip\noindent
1. Let $f(x)$ be an integrable function on $\mathbf{R}$ and let $g(x)$ be the function defined by
\[
g(x)=\int_0^1 tf(x+t)dt
\]
Show that $g$ is continuous on $\mathbf{R}$.
\bigskip\noindent
2. Either prove or give a counterexample to the following statement: given a sequence of measurable functions
$\{f_n(x)\}$ defined on $[0,1]$ converging pointwise to a limit $f(x)$ and a positive integrable function $g(x)$
on $[0,1]$ such that $|f(x)|\leq g(x)$ for all $x\in [0,1]$, then $f(x)$ is integrable and
$\lim_{n\to\infty}\int_0^1 f_n(x)dx = \int_0^1 f(x) dx$.
\bigskip\noindent
3. Suppose the $\{f_c(x)\}_{c\in [a,b]}$ is a family of measurable functions defined on $\mathbf{R}$, and suppose
that for each $x$, $c \rightarrow f_c(x)$ is continuous. Show that $g(x)=\sup\{f_c(x)| c\in [a,b]\}$ is
measurable.
\bigskip\noindent
4. Suppose that $g\in L^\infty([a,b])$ and suppose that $\{f_n\}$ is a sequence of measurable functions converging
to $f$ in measure on $[a,b]$. Show that $gf_n$ converges to $gf$ in measure on $[a,b]$.
\bigskip\noindent
5. Suppose that $f(x)$ is a measurable function on $[0,\infty)$ with the property that $\int_0^\infty |f(x)|^2dx <
\infty$. Show that
\[
\lim_{x\to\infty}x^{\frac{1}{2}}\int_x^\infty\frac{f(t)}{t}dt =0
\]
\bigskip
\noindent Prepared by: Zeljko Cuckovic, Geoffrey Martin\hfill
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