% LaTeX file -- AM/PM 331, Fall 2011, Assignment 5

\documentclass[11pt]{amsart}
\usepackage{amssymb, amstext, amscd, amsmath}
\pagestyle{empty}
%\topmargin 1cm
%\textheight 9in
\textwidth 6in
\oddsidemargin .25in
\evensidemargin .25in
%\voffset -1.5cm

\newcommand{\bN}{{\mathbb N}}
\newcommand{\bQ}{{\mathbb Q}}
\newcommand{\bR}{{\mathbb R}}
\newcommand{\bC}{{\mathbb C}}
\newcommand{\bZ}{{\mathbb Z}}

\newcommand{\Bu}{{\mathbf{u}}}
\newcommand{\Bv}{{\mathbf{v}}}
\newcommand{\Bw}{{\mathbf{w}}}
\newcommand{\Bx}{{\mathbf{x}}}
\newcommand{\By}{{\mathbf{y}}}
\newcommand{\Bz}{{\mathbf{z}}}

\newcommand{\rC}{{\mathrm{C}}}

\newcommand{\FOR}{\text{ for }}
\newcommand{\FORAL}{\text{ for all }}
\newcommand{\AND}{\text{ and }}
\newcommand{\IF}{\text{ if }}
\newcommand{\OR}{\text{ or }}
\newcommand{\qand}{\quad\text{and}\quad}
\newcommand{\qor}{\quad\text{or}\quad}
\newcommand{\qfor}{\quad\text{for}\quad}
\newcommand{\qforal}{\quad\text{for all}\quad}
\newcommand{\qif}{\quad\text{if}\quad}

\newcommand{\ol}{\overline}
\newcommand{\ep}{\varepsilon}
\newcommand{\dlim}{\displaystyle\lim\limits}
\renewcommand{\phi}{\varphi}
\newcommand{\mt}{\emptyset}
\newcommand{\dint}{\displaystyle\int}
\newcommand{\dsum}{\displaystyle\sum}

\newcounter{exerci}[section]
\renewcommand{\theexerci}{\arabic{exerci}.}
\newenvironment{Ex}{\begin{list}%
 {\theexerci\hfill}{\usecounter{exerci}\rightmargin=0pt\leftmargin=12pt%
 \labelwidth=12pt\labelsep=6pt\itemsep=3ex}}{\end{list}\medbreak}
\newenvironment{parts}{\vspace{1ex}\begin{enumerate}\renewcommand{\itemsep}{1ex}
 \renewcommand{\labelenumi}{(\alph{enumi})}}{\end{enumerate}}
\newcommand{\hint}{\textbf{Hint:} }

\begin{document}
\thispagestyle{empty}

%%%%%%%   TITLE   %%%%%%
\begin{center}
\textbf{\large AMath/PMath 331 \\ Assignment 5\\
Due Monday November 7}
\end{center}
\vskip 2em 

\begin{Ex}

\item Prove that $\dsum_{n=1}^\infty \dfrac1{x^2+n^2}$ converges uniformly on the whole real line.

\item  Show that if $f\in \rC[0,1]$, then $\|B_nf\|_\infty \le \|f\|_\infty$ for all $n \ge 0$.\\

\item Look at the proof of the Weierstrass Theorem using the function $f(x) = \big| x-\frac12 \big|$.
For $\ep = \frac12 10^{-3}$, what value of $N$ is required to obtain $\| f - B_Nf \|_\infty < \ep$?\\


\item  Suppose that $f\in \rC^1[0,1]$. Prove that $f$ is a $\rC^1$-limit of polynomials.
i.e. find polynomials $p_n$ so that $\|f-p_n\|_{\rC^1} = \max\{ \|f-p_n\|_\infty,\, \|f'-p_n'\|_\infty \}$
tends to $0$. \hint Approximate $f'$ by polynomials and integrate.

\item Suppose that $f\in \rC[0,1]$ satisfies $\dint_0^1 f(x) x^n \,dx = 0$ for all integers $n\ge0$.\\
Prove that $f=0$. \\
\hint Use polynomial approximation to show that $\dint_0^1 |f(x)|^2 \,dx = 0$.


\item  
\begin{parts}
\item If $a,b,c$ are real numbers, show that the quadratic polynomial $p$ such that
$p(0)=a$, $p(\frac12)=b$ and $p(1)=c$ satisfies $\|p\|_{[0,1]} \le |a|+|b|+|c|$.\\
\hint find three quadratics which take the values $\{0,0,1\}$ on $\{0,\tfrac12,1\}$, and combine.
\item If $f\in \rC[0,1]$, show that there is a sequence of polynomials $p_n$
converging uniformly to $f$ on $[0,1]$ such that $p_n(0)=f(0)$, 
$p_n(\frac12)=f(\frac12)$ and $p_n(1)=f(1)$. \\
\hint start with a sequence of polynomials converging to $f$, and modify it using (a) 
to make the polynomials agree with $f$ at the three points.
\item \textbf{Bonus.} Show that there is a sequence of polynomials $p_n$ converging to $f$
uniformly on $[0,1]$ such that $p_n(\frac{k}{n!}) = f(\frac{k}{n!})$ for $0 \le k \le n!$.\\
\end{parts}

\item Find the closest cubic polynomial to $|x|$ on $[-1,1]$. How close is it?


\end{Ex}
\end{document}