% LaTeX file -- PM 450 Winter 2017, Assignment 2

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\begin{document}
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%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 450 \hfill Assignment 2 \hfill Due Wednesday February 1.}} 
\vskip 2em 

\begin{Ex}

\item Convert the Laplacian $\Delta u = u_{rr} + \frac1r u_r + \frac1{r^2} u_{\theta\theta}$ to Cartesion coordinates.\\
\hint Compute $u_r,\ u_{rr},\ u_\theta$ and $u_{\theta\theta}$ in terms of partials w.r.t. $x$ and $y$.

\item Let $f(\theta) = \theta$ for $-\pi < \theta \le \pi$.
\begin{parts}
\item Compute the Fourier series of $f$, and convert it to a sum of sines and cosines.
\item Does this series converge uniformly on $[-\pi,\pi]$?
\item Evaluate $\dlim_{r\to 1^-} u(r,\pi)$.
\end{parts}

\item Prove that if $\{A_n\}_{n\in\bZ}$ is a bounded sequence, then $u(r,\theta) = \sum_{n\in\bZ} A_n r^{|n|} e^{in\theta}$ is $C^\infty$ on the open disc $\bD$, i.e. show that all partial derivatives $\tfrac{\partial^{j+k}}{\partial r^j\partial \theta^k} u$ exist and are continuous.

\item Show that every Riemann integrable function $f$ on $\bT$ is a limit of continuous functions in the $L^2(\bT)$ norm.
\hint Assume that $f$ is real valued first. Find a partition for which the upper and lower Riemann sums of $\|f\|_2^2 = \dfrac1{2\pi}\dint_{-\pi}^\pi |f(\theta)|^2\,d\theta$ are close. Use this partitition to define a continuous function $g$ that fits within the same upper and lower bounds. Estimate $\|f-g\|_2^2$.

\item Let $f$ be a \textit{positive} Riemann integrable
$2\pi$-periodic function with harmonic extension $u(r,\theta)$.
\begin{parts}
\item Prove that $u(r,\theta) \ge 0$.
\item Prove that $\dfrac{1-r}{1+r} \le P(r,\theta) \le \dfrac{1+r}{1-r}$.
\item Hence prove that $\Big(\dfrac{1-r}{1+r}\Big)u(0,0) \le u(r,\theta) \le \Big(\dfrac{1+r}{1-r}\Big) u(0,0) .$
\end{parts}

\item 
\begin{parts}
\item Prove that $\dfrac1{2\pi}\dint_{-\pi}^\pi P(r,\theta - t) P(s, t) \,dt = P(rs, \theta)$.\\
\hint Use the series expansion of the Poisson kernel. Be careful about convergence issues.

\item Let $f(\theta)$ be a Riemann integrable $2\pi$-periodic function, and let $u(r,\theta)$ be its harmonic extension.
Fix $s \in (0,1)$, and define $g(\theta) = u(s,\theta)$. Prove that the harmonic extension of $g$ is $u(rs, \theta)$.
\end{parts}

\item Say that a $2\pi$-periodic function $f$ is in $C^k(\bT)$ if $f$ has derivatives up to order $k$ and all are continuous and $2\pi$-periodic.
\begin{parts}
\item Show if $f \in C^1(\bT)$, then $|\hat f(n)| \le C/|n|$ for some constant $C$.  \qquad
\hint Integrate by parts.
\item Show by induction that if $f \in C^k(\bT)$, then $|\hat f(n)| \le Cn^{-k}$ for some constant $C$.
\item In particular, if $f\in C^2(\bT)$, show that the Fourier series for $f$ converges uniformly on $\bT$ to a continuous function.
\item Conversely show that if $f \in C(\bT)$ satisfies  $|\hat f(n)| \le Cn^{-k}$ for an integer $k \ge 2$,
then $f \in C^{k-2}(\bT)$. \qquad\hint Term-by-term differentiation and the $M$-test.
\item Hence give necessary and sufficient conditions on $\hat f$ for $f$ to be in $C^\infty(\bT)$.
\end{parts}


\end{Ex}
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