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\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}