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\begin{document}
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\vspace*{-.7in}

\noindent
\textbf{Math 5616H} \hfill \textbf{Homework 4}

\noindent
\textbf{Posted: 2/14; Updated: 02/19; Due: Friday, 2/20/2015}\\
The problem set is due at the beginning of the class on
Friday. \smallskip

\noindent
\textbf{Reading}: 7.3-6, 7.16-25. \smallskip

\begin{prob}
  Define a sequence of polynomials $P_0, P_1, P_2, \dots$ by $P_0(x) =
  0$ and
\[
P_{n+1}(x) = P_n(x) + \frac{x^2 - (P_n(x))^2}{2}, \qquad n = 0, 1, 2,
\dots
\]
Prove that the sequence converges uniformly to the function $\abs{x}$
on the interval $[-1,1]$. \emph{Hint}: Note that all $P_n$'s are even
and consider the interval $[0,1]$. Then apply Dini's theorem (7.13).
\end{prob}

\begin{prob}
  Consider the following functional series:
\[
e^{-x} + 2 e^{-2x} + \dots + n e^{-nx} + \dots,
\]
which we understand as a sequence of partial sums. Show that the
series converges pointwise on $(0, +\infty)$ to a continuous function
$f(x)$. Compute $\int_{\log 2}^{\log 3} f(x) dx$. \emph{Hint}: For
continuity use the material on power series we studied last term. For
computation of the integral, use Dini's theorem (7.13) and the theorem
on passing to limit in an integral.
\end{prob}

\begin{prob}
  Prove that the following functional series:
\[
\frac{\sin 2 \pi x}{2} + \frac{\sin 4 \pi x}{4} + \dots + \frac{\sin
  2^n \pi x}{2^n} + \dots
\]
converges uniformly on $(-\infty, +\infty)$. Show that it cannot be
differentiated termwise on any interval $[a,b]$.
\end{prob}

\begin{prob}
Show that
\[
\frac{1}{1+x} + \frac{2 x}{1+x^2} + \dots + \frac{m x^{m-1}}{1+x^m} +
\dots = \frac{1}{1-x},
\]
where $m = 2^{n-1}$ and $-1 < x < 1$, in the sense of pointwise
convergence.
\end{prob}

\begin{prob}
  Suppose $\F = \{f_i \; | \; i \in I\}$ be a finite collection of
  real- or complex-valued functions on a metric space $X$,
  \emph{i.e}., the set $I$ is finite. Show that $\F$ is equicontinuous
  if and only if all the $f_i$'s are uniformly continuous.
\end{prob}

\begin{prob}
Consider the infinite sequence of functions
\[
f_n(x) = \frac{x}{x + \frac{1}{n}} , \qquad x \in [0, 1], n \in \nn.
\]
Show that each function $f_n$ is uniformly continuous, but the
collection is not equicontinuous by showing that it has no uniformly
convergent subsequence and applying the Arzel\` a-Ascoli theorem
(7.25).
\end{prob}

\begin{prob}
  Consider and infinite collection $\F = \{f_n\; | \; n \in \nn\}$ of
  functions
\[
f_n(x) = e^{-n(x-n)^2} , \qquad x \in \nr.
\]
\begin{enumerate}
\item Show that $\F$ is closed in $\B\C (\nr, \nr)$ but not closed in
  $\C([-a,a], \nr)$ for any $a > 0$. \emph{Hint}: To show closedness
  over $\nr$, estimate the distance between $f_n$ and $f_m$ by a
  constant, say, $1/2$. This would imply that this set is discrete,
  i.e., each point of it being open, thereby making all subsets open
  and closed.
\item Show that this collection is not equicontinuous on $\nr$ but is
  equicontinuous on an interval $[-a,a]$ for any $a > 0$.
\end{enumerate}
\end{prob}

\end{document}