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\title{Analysis II}
\author{Homework 4}
\date{Due in class Monday, March 4, 2019}
\maketitle

\begin{enumerate}


\item Let $f \colon \R \to \R$ be an arbitrary function.  Consider the
sequence of left translates $f_n(x) = f (x + n)$, where $n \geq 1$.
Prove that $f_n$ converges uniformly on $[0, \infty)$ to the constant
function $c$ if and only if $\lim_{x\to\infty} f(x) = c$.


\item Test the following series for uniform convergence on $\R$:
\begin{enumerate}
\item $\displaystyle \sum_{n=1}^\infty \frac{x}{(n+x^2)^2}$
\item $\displaystyle \sum_{n=1}^\infty \frac{\sin(nx)}n$
\end{enumerate}


\item Let $f(x,y) = y^3 \, \e^{-y^2 x}$ and define
\[
  \phi(y) = \int_0^\infty f(x,y) \, \d x \,.
\]
\begin{enumerate}
\item Show that, at $y=0$,
\[
  \phi'(y) \neq \int_0^\infty \frac{\partial f}{\partial y}(x,y) \, \d
  x \,.
\]
\item Which condition in the theorem from class on the interchange of
differentiation and integration is violated?
\end{enumerate}

\item Which of the following sequences of functions is equicontinuous
on $\R$?  Prove or disprove: (a) $f_n(x) = \sin(nx)$, (b)
$f_n(x) = {\sin(nx)}/n$.

\item (Rudin, Exercise 7.16.) Suppose $\{ f_n\}$ is an equicontinuous
sequence of functions on a compact set $K$, and $\{f_n\}$ converges
pointwise on $K$.  Prove that $\{f_n\}$ converges uniformly on $K$.

\item (Rudin, Exercise 7.18.) Let $\{f_n\}$ be a uniformly bounded
sequence of functions which are Riemann-integrable on some interval
$[a,b]$ and define, for $x \in [a,b]$, 
\[
  F_n(x) = \int_a^x f_n(t) \, \d t \,.
\]
Prove that there exists a subsequence $\{ F_{n_k} \}$ which converges
uniformly on $[a,b]$.

\end{enumerate}





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