% LaTeX file -- PM 450 Winter 2017, Assignment 3

\documentclass[11pt]{amsart}
\usepackage{amssymb, amstext, amscd, amsmath}
%\usepackage{fullpage}
\pagestyle{empty}
\topmargin 0in
\textheight 9.0in
\textwidth 6.5in
\oddsidemargin .0in

\voffset -8mm

%      Blackboard bold letters
\newcommand{\bA}{{\mathbb{A}}}
\newcommand{\bB}{{\mathbb{B}}}
\newcommand{\bC}{{\mathbb{C}}}
\newcommand{\bD}{{\mathbb{D}}}
\newcommand{\bE}{{\mathbb{E}}}
\newcommand{\bF}{{\mathbb{F}}}
\newcommand{\bG}{{\mathbb{G}}}
\newcommand{\bH}{{\mathbb{H}}}
\newcommand{\bI}{{\mathbb{I}}}
\newcommand{\bJ}{{\mathbb{J}}}
\newcommand{\bK}{{\mathbb{K}}}
\newcommand{\bL}{{\mathbb{L}}}
\newcommand{\bM}{{\mathbb{M}}}
\newcommand{\bN}{{\mathbb{N}}}
\newcommand{\bO}{{\mathbb{O}}}
\newcommand{\bP}{{\mathbb{P}}}
\newcommand{\bQ}{{\mathbb{Q}}}
\newcommand{\bR}{{\mathbb{R}}}
\newcommand{\bS}{{\mathbb{S}}}
\newcommand{\bT}{{\mathbb{T}}}
\newcommand{\bU}{{\mathbb{U}}}
\newcommand{\bV}{{\mathbb{V}}}
\newcommand{\bW}{{\mathbb{W}}}
\newcommand{\bX}{{\mathbb{X}}}
\newcommand{\bY}{{\mathbb{Y}}}
\newcommand{\bZ}{{\mathbb{Z}}}

%      Lower case bold letters
\newcommand{\Ba}{{\mathbf{a}}}
\newcommand{\Bb}{{\mathbf{b}}}
\newcommand{\Bc}{{\mathbf{c}}}
\newcommand{\Bd}{{\mathbf{d}}}
\newcommand{\Be}{{\mathbf{e}}}
\newcommand{\Bf}{{\mathbf{f}}}
\newcommand{\Bg}{{\mathbf{g}}}
\newcommand{\Bh}{{\mathbf{h}}}
\newcommand{\Bi}{{\mathbf{i}}}
\newcommand{\Bj}{{\mathbf{j}}}
\newcommand{\Bk}{{\mathbf{k}}}
\newcommand{\Bl}{{\mathbf{l}}}
\newcommand{\Bm}{{\mathbf{m}}}
\newcommand{\Bn}{{\mathbf{n}}}
\newcommand{\Bo}{{\mathbf{o}}}
\newcommand{\Bp}{{\mathbf{p}}}
\newcommand{\Bq}{{\mathbf{q}}}
\newcommand{\Br}{{\mathbf{r}}}
\newcommand{\Bs}{{\mathbf{s}}}
\newcommand{\Bt}{{\mathbf{t}}}
\newcommand{\Bu}{{\mathbf{u}}}
\newcommand{\Bv}{{\mathbf{v}}}
\newcommand{\Bw}{{\mathbf{w}}}
\newcommand{\Bx}{{\mathbf{x}}}
\newcommand{\By}{{\mathbf{y}}}
\newcommand{\Bz}{{\mathbf{z}}}

%      Capital script letters
  \newcommand{\A}{{\mathcal{A}}}
  \newcommand{\B}{{\mathcal{B}}}
  \newcommand{\C}{{\mathcal{C}}}
  \newcommand{\D}{{\mathcal{D}}}
  \newcommand{\E}{{\mathcal{E}}}
  \newcommand{\F}{{\mathcal{F}}}
  \newcommand{\G}{{\mathcal{G}}}
\renewcommand{\H}{{\mathcal{H}}}
  \newcommand{\I}{{\mathcal{I}}}
  \newcommand{\J}{{\mathcal{J}}}
  \newcommand{\K}{{\mathcal{K}}}  
\renewcommand{\L}{{\mathcal{L}}}
  \newcommand{\M}{{\mathcal{M}}}
  \newcommand{\N}{{\mathcal{N}}}
\renewcommand{\O}{{\mathcal{O}}}
\renewcommand{\P}{{\mathcal{P}}}
  \newcommand{\Q}{{\mathcal{Q}}}
  \newcommand{\R}{{\mathcal{R}}}
\renewcommand{\S}{{\mathcal{S}}}
  \newcommand{\T}{{\mathcal{T}}}
  \newcommand{\U}{{\mathcal{U}}}
  \newcommand{\V}{{\mathcal{V}}}
  \newcommand{\W}{{\mathcal{W}}}
  \newcommand{\X}{{\mathcal{X}}}
  \newcommand{\Y}{{\mathcal{Y}}}
  \newcommand{\Z}{{\mathcal{Z}}}

\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{\conv}{\operatorname{conv}}
\newcommand{\dint}{\displaystyle\int}
\newcommand{\dsum}{\displaystyle\sum}
\newcommand{\Dim}{\operatorname{dim}}
\newcommand{\dist}{\operatorname{dist}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\ext}{\operatorname{ext}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\im}{\operatorname{Im}}
\newcommand{\ind}{\operatorname{ind}}
\newcommand{\ol}{\overline}
\newcommand{\ran}{\operatorname{Ran}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\re}{\operatorname{Re}}
\newcommand{\RI}{\operatorname{RI}}
\newcommand{\sgn}{\operatorname{sgn}}
\newcommand{\spn}{\operatorname{span}}
\newcommand{\spr}{\operatorname{spr}}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\ep}{\varepsilon}

\newcommand{\CT}{\mathrm{C}(\bT)}
\newcommand{\rC}{\mathrm{C}}
\newcommand{\dlim}{\displaystyle\lim\limits}
\newcommand{\one}{\boldsymbol{1}}
\renewcommand{\phi}{\varphi}
\newcommand{\upchi}{{\raise.35ex\hbox{$\chi$}}}
\newenvironment{sbmatrix}{\left[\begin{smallmatrix}}{\end{smallmatrix}\right]}
\newcommand{\mt}{\emptyset}
\newcommand{\lip}{\langle}
\newcommand{\rip}{\rangle}
\newcommand{\ip}[1]{\lip #1 \rip}
\newcommand{\wh}{\widehat}
\newcommand{\sot}{\textsc{sot}}
\newcommand{\wot}{\textsc{wot}}

\newcounter{exerci}[section]
\renewcommand{\theexerci}{\arabic{exerci}.}
\newenvironment{Ex}{\begin{list}%
 {\theexerci\hfill}{\usecounter{exerci}\rightmargin=0pt\leftmargin=12pt%
 \labelwidth=12pt\labelsep=6pt\itemsep=2ex}}{\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   %%%%%%
{\noindent\textbf{\large PM 450 \hfill Assignment 3 \hfill Due Wednesday February 15.}} 
\vskip 2em 

\begin{Ex}

\item Define $V_n(\theta) = \dsum_{k=-n-1}^{n+1} e^{ik\theta} + \dsum_{k=n+2}^{2n+1} \tfrac{2n+2-k}{n+1}  \big( e^{ik\theta} + e^{-ik\theta}\big)$.
\begin{parts}
\item Prove that $V_n(\theta) = 2 K_{2n+1}(\theta) - K_n(\theta)$  (where $K_n$ is the F\'ejer kernel).
\item Hence prove that $V_n$ is an even summability kernel.
\end{parts}


\item Let $f(\theta) = \theta^3 - \pi^2\theta$ on $[-\pi,\pi]$.
\begin{parts}
\item Compute the Fourier series for $f$.
\item What can be said about the convergence of this series? 
\item Evaluate at $\theta=\pi/2$ to evaluate $\dsum_{n=0}^\infty \dfrac{(-1)^n}{(2n+1)^3}$.
\end{parts}


\item Let $f(\theta) = 
\begin{cases}
 1 &\qif 0 \le |\theta| \le \frac{2\pi}3 \\
 0 &\qif \frac{2\pi}3 \le |\theta| \le \pi
\end{cases}.$
\begin{parts}
\item Compute the Fourier series for $f$.
\item What can be said about the convergence of this series? 
\item Evaluate at $\theta=0$ and hence compute $\dsum_{n=0}^\infty \dfrac1{(3n+1)(3n+2)}$.
\end{parts}


\item For $f\in \RI(\bT)$, and define the translation of $f$ by $t$ be $f_t(\theta) = f(\theta-t)$.
\begin{parts}
\item Prove that $\dlim_{t\to 0} \|f-f_t\|_2 = 0$.
\hint First prove it for $g \in \rC(\bT)$, and then approximate $f$ by continuous functions in the $L^2$ norm.
\item Prove that if $f,g \in \RI(\bT)$, then $f*g$ is continuous.
\hint Cauchy-Schwarz inequality for the $L^2$ inner product.
\end{parts}
 

\item Let $p(\theta) = \sum_{k=-n}^n a_k e^{ik\theta}$ be a trig polynomial of degree $n$.
\begin{parts}
\item Compute the Fourier series of $g_n(\theta) = -2nK_{n-1}(\theta)\sin(n\theta)$.
\item Hence obtain a formula for $p'(\theta)$ as a convolution.
\item Prove that $\|p'\|_\infty \le 2n \|p\|_\infty$.
\end{parts}


\item  Suppose that $f(\theta)$ is a monotone increasing real valued function on $[-\pi,\pi]$.
\begin{parts}
\item Show that if $\{k_n\}$ is an even summability kernel, then $f*k_n$ converges pointwise, and find the limit.
\item Prove that $|\hat f(n)| \le C/|n|$ for $n\ne 0$.\\
\hint WLOG $f(-\pi)=0$ (why?). Prove it first when $f$ is a (monotone) step function by integrating over each interval and rearranging the sum. Then approximate a general monotone $f$.
\item Hence show that the Fourier series of $f$ converges at every point in $[-\pi,\pi]$.
\end{parts}



\end{Ex}
\end{document}