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\title{Math 581 Assignment 4}
\author{Due Friday March 2}
\date{Winter 2012} % Activate to display a given date or no date
\begin{document}
\maketitle
\begin{enumerate}[1.]
\item
Let $\omega\subset\R$ be an open interval, and $u\in\tstD'(\omega)$. Let $0\leq k\leq \infty$.
\begin{enumerate}[a)]
\item
Show that if $u'=0$ then $u$ is a constant function.
\item
Show that if $u'=f$ with $f\in C^{k}(\omega)$, then $u\in C^{k+1}(\omega)$.
\item
Show that if $u'+au=f$ with $a\in C^{\infty}(\omega)$ and $f\in C^{k}(\omega)$, then $u\in C^{k+1}(\omega)$.
\item
Extend the above result to $n$-th order linear ODE's with smooth coefficients, assuming that the leading order coefficient does not vanish on $\omega$.
This would prove in particular hypoellipticity of linear ordinary differential operators with smooth coefficients (provided the leading order coefficient is nowhere zero).
One can also prove analytic-hypoellipticity of such operators with analytic coefficients.
\end{enumerate}
\item
A distribution $u$ is called {\em nonnegative} if $u(\varphi)$ is nonnegative for every nonnegative test function $\varphi$.
Show that a distribution is nonnegative if and only if it is a nonnegative Radon measure.
Note that this means any Jordan-type decomposition would fail for distributions:
Radon measures are the only distributions which can be written as the difference of two nonnegative distributions.
\item
Let us define the Fourier transform by
$$
\hat{u}(\xi) = \alpha\int e^{i\beta x\cdot\xi} u(x) \exd x,
$$
for $u\in\tstS(\R^n)$, where $\alpha,\beta\in\R$ are constants.
Derive a formula for the inverse transformation.
List some common and/or convenient choices for the constants $\alpha$ and $\beta$.
For $u,v\in\tstS$, prove (or derive a formula for) the followings.
\begin{enumerate}[a)]
\item
Parseval's formula: $\int u\bar{v} = \gamma \int \hat u \bar{\hat v}$, where $\gamma=\gamma(\alpha,\beta)$ is a constant.
\item
$\widehat{u*v} = \hat u \hat v$.
\item
$\widehat{uv}=\gamma \hat u * \hat v $.
\item
Derive a formula for $\widehat{u\circ A}$, where $A$ is an $n\times n$ invertible matrix.
\end{enumerate}
\item
There are (at least) two ways to define the Fourier transform on $L^2(\R^n)$.
\begin{itemize}
\item
Extend the Fourier transform from $\tstS$ to $L^2$ by using the density of $\tstS$ in $L^2$ (as well as the Plancherel bound).
\item
First define the Fourier transform on $\tstS'$ by duality, and then restrict it to $L^2$.
\end{itemize}
Show that these two approaches are consistent with each other.
\item
Give an example of $u\in C(\R^n)$ such that $\varphi\mapsto\int u\varphi$ is a tempered distribution and that there is no polynomial $p$ satisfying
$|u(x)|\leq|p(x)|$ for all $x\in\R^n$.
\item
For each of the following functions,
determine if it is a tempered distribution,
and if so compute its Fourier transform.
a) $\sin x$,
b) $e^{i|x|^2}$,
c) The Heaviside step function,
d) The sign function,
e) $|x|^s$, where $s$ is a real number.
\item
a) Let $a\in\mathscr{E}(\R^n)$.
Prove that the pointwise multiplication $u\mapsto au:\tstS'\to\tstS'$ is well-defined and continuous if and only if for every multi-index $\alpha$ there is a polynomial $p$ such that
$|\partial^\alpha a(x)|\leq p(x)$, $x\in\R^n$.
b) Let $p$ be a polynomial satisfying $|p(i\xi)|\geq c(1+|\xi|)^m$ for all $\xi\in\R^n$, with some constants $c>0$ and $m$.
Operators $p(\partial)$ with $p$ satisfying this condition are called {\em strictly elliptic}.
Show that the equation $p(\partial)u=f$ has a solution for each $f\in\tstS'$.
\end{enumerate}
\end{document}