% LaTeX file -- PM 450 Winter 2017, Assignment 6

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

%\topmargin 1cm
%\textheight 7.8in
%\textwidth 6in
%\oddsidemargin .25in
%\evensidemargin .25in
\voffset -11mm

%      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}

\renewcommand{\ae}{\operatorname{a.e.}}
\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{\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}}
\newcommand{\bsl}{\setminus}
\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=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   %%%%%%
{\noindent\textbf{\large PM 450 \hfill Assignment 6 \hfill Due Wednesday March 29.}} 
\vskip 1.5em 

\begin{Ex}

\item \begin{parts}
\item Using $f(\theta) = \theta^3-\pi^2\theta$ from Assignment 2, Q3, evaluate $\dsum_{n=1}^\infty \dfrac1{n^6}$.\vspace{-3ex}
\item Integrate $f$ and use it to evaluate $\dsum_{n=1}^\infty \dfrac1{n^8}$.
\end{parts}
 

\item Prove that $\{ 1, \sqrt2 \cos n\theta : n \ge1\}$ is an orthonormal basis for $L^2(0,\pi)$ with the inner product
\[ \ip{f,g} = \dfrac1\pi \dint_{(0,\pi)} f \ol{g} .\]

\item Let $a \in \bR\bsl\bZ$. Let $f(\theta) = e^{ia\theta}$ for $\theta\in (-\pi,\pi]$.
Evaluate $\|f\|_2^2$ in two ways and deduce that 
\[ \dsum_{n=-\infty}^\infty \dfrac1{(a-n)^2} = \dfrac{\pi^2}{\sin^2 a\pi} .\]

\item 
\begin{parts}
\item Show that if $f$ is a $2\pi$-periodic $C^1$ function and $\hat f(0)=0$, then $\|f\|_2 \le \|f'\|_2$.
\item Show that if $f$ is a $2\pi$-periodic $C^2$ function, then $\|f'\|_2^2 \le \|f\|_2\,\|f''\|_2$. 
%\item Show that if $f,g \in L^2(\bT)$, then $\dsum_{-\infty}^\infty |\widehat{f*g}(n)| < \infty$. \\
%Hence show that $f*g$ agrees with a continuous function a.e.
\end{parts}


\item Let $1 \le p < \infty$ and suppose that $\frac1p + \frac1q = 1$.
\begin{parts}
\item Prove that if $f, f_n\in L^p(X)$ and $g\in L^q(X)$ and $\dlim_{n\to\infty} \|f-f_n\|_p=0$, then
\[ \lim_{n\to\infty} \int_X f_n g = \int_X fg .\]
\item Show that if $f\in L^p(\bT)$ and $g\in L^q(\bT)$, then
\[ \frac1{2\pi} \int_\bT fg = \lim_{n\to\infty} \sum_{k=-n}^n \big( 1 - \tfrac{|k|}{n+1} \big) \hat f(k) \hat g(-k) .\]
\end{parts}

\item Let $1 \le p < \infty$ and suppose that $\frac1p + \frac1q = 1$. 
Let $f\in L^p(\bT)$ and $g\in L^q(\bT)$. Prove that
\[ \lim_{n\to\infty} \frac1{2\pi} \int_{-\pi}^\pi f(\theta) g(n\theta) \,d\theta = \hat f(0) \hat g(0) .\] 
\hint prove it first when $f$ is a trigonometric polynomial.


\item Fix $1 \le p < \infty$. Define a linear map $S_{p,n} : L^p(\bT) \to L^p(\bT)$ given by $S_{p,n}f = s_n(f) = f*D_n$.
\begin{parts}
\item Prove that the following  statements are equivalent:
\begin{enumerate}
 \renewcommand{\labelenumii}{(\arabic{enumii})}
\item $\sup_{n\ge1} \|S_{p,n}\| < \infty$.
\item $\sup_{n\ge1} \|S_{p,n} (f)\|_p < \infty$ for every $f\in L^p(\bT)$.
\item $S_{p,n}f$ converges to $f$ in the $L^p(\bT)$ norm for every $f\in L^p(\bT)$.
\end{enumerate}
\item Show that $\|S_{1,n}\|=\|D_n\|_1$. Hence show that there is a function $f \in L^1$  such that $s_n(f)$ diverges.
\qquad \hint look at $S_{1,n}(K_m)$ as $m\to\infty$.
\end{parts}


\end{Ex}
\end{document}