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18.104
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Spring 2013
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\begin{center}\textbf{Homework 11}\end{center}

\begin{center}Due April 9th by the beginning of class.\end{center}

\textbf{Problem:} Prove that if $u \in \mathscr S'(\mathbb R^n)$ and $f \in \mathscr S(\mathbb R^n)$ then $u *f$ can be defined as in the remark following \cite[Theorem 8.4.2]{frjo}. Show that given  $u \in \mathscr S'(\mathbb R^n)$, there is $f \in \mathscr S(\mathbb R^n)$ such that $u*f=u$ if and only if $\hat u \in \mathscr E'(\mathbb R^n)$. (This shows that regularization theorems like \cite[Theorem 5.2.1]{frjo} are trickier when both the function and distribution have noncompact support.)

\textit{Hint:} Solving $u*f=u$ is equivalent to solving $\hat u \hat f = \hat u$. Use $\lim_{|\xi| \to \infty} \hat f (\xi) = 0$ to prove one implication, and make $\hat f = 1$ near the support of $\hat u$ for the other.

\textbf{Solution:}


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