Math 6410 • ODE

Let \(\Omega\subset\R\) be open, and let \(C_0^\infty(\Omega)\) be the space of
\(C^\infty\) functions with support in \(\Omega\). Define convergence of a sequence
\(\varphi_n\) as follows: \(\varphi_n\to0\) iff \(\exists\) a compact set \(K\subset\Omega\)
such that
\begin{enumerate}
\item[(i)] Support \(\varphi_n\subset K\) for all \(n\)
\item[(ii)] \(\displaystyle\lim_{n\to\infty}\sup_{x\in K}\lvert\varphi_n^{(p)}(x)\rvert=0\) for
\(p\in Z^+\).
\end{enumerate}
\begin{enumerate}
\item Define what is meant by a distribution on \(C_0^\infty(\Omega)\).
\item Define
\[
  \mu_f(\varphi)=\int_{-\infty}^{\infty} f(x)\bar{\varphi}(x)\,dx
\]
where \(f\in L^1_{\mathrm{loc}}(\Omega)\) and \(\varphi\in C_0^\infty(\Omega)\). Does this
define a distribution on \(C_0^\infty(\Omega)\)? Give reasons.
\item Give a concrete example of a regular distribution and verify that is is a
distribution.
\item Show how to define the derivatives of a regular distribution.
\end{enumerate}