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18.104
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Spring 2013
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\begin{center}\textbf{Homework 9}\end{center}

\begin{center}Due March 14th by the beginning of class.\end{center}

\textbf{Problem:} Let $a>0$ and let $1_{(0,a)}\colon \mathbb R \to \{0,1\}$ be the characteristic function of $(0,a)$; this is the function which is $1$ on $(0,a)$ and $0$ elsewhere. Find $u_a \in \mathscr D'(\mathbb R)$ such that $\textrm{supp }u_a \subset [0,\infty)$ and $1_{(0,a)} * u_a = \delta$.

\textit{Hint:} Try $u_a = \sum_{k=0}^\infty \partial \delta_{ka}$. Use the results of \cite[page 52]{frjo} to simplify $1_{(0,a)} * u_a$ (justifying carefully the interchange of sum and convolution).

\textbf{Solution:}


\begin{thebibliography}{00}
\bibitem[FrJo]{frjo} G. Friedlander and M. Joshi. The Theory of Distributions, second edition, Cambridge University Press, 1998.
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