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\begin{document}
\title{Homework 5: Discrete Sobolev spaces}
\date{Assigned Fri 05/04. Due Wed 05/09}
\maketitle

Since you already had a (possibly) traumatic midterm to deal with this week, this homework is short.
\begin{questions}
\item Let $L:C^2 \to C^0$ be defined by
$$Lu = \sum_{i,j=1}^d a_{ij}\frac{\partial^2u}{\partial x_i \partial x_j} + \sum_{i=1}^d b_i \frac{\partial u}{\partial x_i} + c u$$
where $a_{ij}$, $b_i$ and $c$ are (given) constants, and $a_{ij} = a_{ji}$. The operator $L$ is called \textit{elliptic} if there exists $\lambda > 0$ such that for all $\xi = (\xi_1, \dots, \xi_d) \in \R^d$, we have $\abs{\sum_{i,j} a_{ij} \xi_i \xi_j} \geq \lambda \abs{\xi}^2$. If $L$ is elliptic, $L u = g$ and $g \in H^s$, show that $u \in H^{s+2}$. Conclude that if $g \in C^\infty$, then $u \in C^\infty$. \noteb{Note that $L u = \triangle u$ is elliptic, and hence what we did in class is a special case of the above. This is usually know as elliptic regularity.}

\item Pick $0\leq s < t$. Let $S = \{ f \in H^t \st \norm{f}_{H^t} \leq 1 \}$. Show that $\bar{S} \subset H^s$ is compact. \noteb{This is a usual feature in functional analysis. When you are working with a space of functions, the method by which you obtain compact sets is to impose some restriction on the derivatives! This is something you saw on your midterm too with $C^\alpha$. The proof here is not as horrific as the midterm question. Use something similar to the last problem on the first assignment\dots.}

\items Let $s > \frac{d}{2}$. Suppose $\alpha = s - \frac{d}{2} \in (0,1)$, and $f \in H^s([0,1]^d)$, then show that $f \in C^\alpha([0,1]^d)$.

\items \textit{(Poincar\'e Inequality)} Say $f \in C^1[(-\pi, \pi)]$ and $\int_{-\pi}^\pi f = 0$. Show that $\Lpnorm{f}{2} \leq \Lpnorm{f'}{2}$. \noteb{If this is too easy, try proving it without Fourier Series! In this case you will only be able to show $\exists c \suchthat \Lpnorm{f}{2} \leq c \Lpnorm{f'}{2}$.}
\end{questions}\medskip


\title{Midterm}
\date{Assigned Thu 05/03 2:15PM. Due Fri 05/04 2:15PM.}
\maketitle

If you have trouble with 3(b), or 4(b) don't lose sleep over it. Come back to them if you have time: 3(b) is for the most part straightforward (if you follow the hint), but has some pretty horrific details. 4(b) is short and elegant, however if you don't see the proof I have in mind, it is likely you will spew some 300 pages and get no where near the correct solution.

\begin{questions}

\item Let $X$ be an inner product space. Suppose $\{e_n | n \in \N \}$ is an orthonormal set such that for any $x \in X$, $\norm{x}^2 = \sum \abs{\ip{x}{e_n}}^2$. Show that for any $x, y \in X$, $\ip{x}{y} = \sum \ip{x}{e_n} \overline{\ip{y}{e_n}}$.

\item Let $X = \{f \st f:[0,1] \to \C \text{ is continuous}\}$. For $f, g \in X$, define $\ip{f}{g} = \int_0^1 f \bar{g}$. I've mentioned before that $X$ is an inner product space, but not Hilbert (in fact we defined $L^2$ to be the completion of $X$). Give an explicit example of a Cauchy sequence in $X$ which is not convergent.

\item Let $\alpha \in (0,1)$. Let $C^\alpha([0,1])$ be the set of all continuous functions $f:[0,1] \to \R$ such that $\sup \{ \frac{\abs{f(x) - f(y)}}{\abs{x - y}^\alpha} \st x, y \in [0,1] \text{ and } x \neq y \} < \infty$.
\begin{parts}
\item If $f \in C^\alpha$, we define $\norm{f}_{C^\alpha} = \Lpnorm{f}{\infty} + \sup\{ \frac{\abs{f(x) - f(y)}}{\abs{x - y}^\alpha} \st x, y \in [0,1] \text{ and } x \neq y \}$. Show that $C^\alpha$ with norm $\norm{f}_{C^\alpha}$ is a Banach space. \noteb{In interest of saving trees, only check that the triangle inequality is satisfied, and that Cauchy sequences converge.}
\item Pick $\beta > \alpha$, and let $S = \{ f \in C^\alpha \st \norm{f}_{C^\beta} \leq 1 \}$. Show that $S$ is a compact subset of $C^\alpha$. \hintb[Note]{You need $\beta > \alpha$. $S$ is not a compact subset of $C^\beta$.}\\
\hint{Let $C_n$ be the set of all piecewise linear functions such that $\forall k \in \{0, \dots, n\}$, $f(\frac{k}{n}) \in \{ 0, \frac{\pm 1}{n}, \dots, \frac{\pm n}{n} \}$. Note $C_n$ is finite. Show that for any $\epsilon > 0$, there exists $n$ such that $S \subset \cup_{f\in C_n} B_\epsilon(f)$. This implies total boundedness of $S$.}
\end{parts}

\item Let $(\phi_n)$ be an approximate identity.
\begin{parts}
\item For any $k \in \Z$, show that $\dlim_{n \to \infty} \hat\phi_n(k)$ exists.
\item Let $F:\Lpspace{1} \to \lpspace{\infty}$ be defined by $F(f) = (\hat{f}(n))_n$ (as in Homework 4 problem 6). Show that $(F(\phi_n))_n$ does not converge in $\lpspace{\infty}$.
\end{parts}
\end{questions}

\end{document}