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\begin{document}

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\section*{Colombeau algebra}

\hypertarget{content}{}\section*{{Content}}\label{content}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The [[product of distributions]] is not canonically defined on all distributions; it is only partially defined on pairs of distributions whose sum of [[wave front sets]] does not intersect zero (``[[microlocal analysis]]'').

In contrast, \hyperlink{Colombeau92}{Colombeau 92} defines algebra structures on distributions, now called \emph{Colombeau algebras} (see \hyperlink{Gratus13}{Gratus 13} for review) which are globally defined, but at the cost that

\begin{enumerate}%
\item they are not uniquely defined,


\item they do not restrict to the usual pointwise product on all functions.



\end{enumerate}
Briefly: Colombeau considers sequences of functions that converge to distributions (weakly) and defines the product of two distributions as the product of the sequences. This product is not independent of the chosen sequences, which means that the level of abstraction achieved by distribution theory is abandoned.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Jean Fran\c{c}ois Colombeau: \emph{Multiplication of distributions. A tool in mathematics, numerical engineering and theoretical physics.}, Springer 1992 (\href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0815.35002&format=complete}{ZMATH entry})


\item Jonathan Gratus, \emph{Colombeau Algebra: A pedagogical introduction} (\href{https://arxiv.org/abs/1308.0257}{arViv:1308.0257})


\item Hans Vernaeve, \emph{Algebras of generalized functions and Nonstandard analysis}, 2008 (\href{http://fibonacci.dm.unipi.it/cluster-pages/ultramath/slides/vernaeve-slides.pdf}{pdf})


\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Colombeau_algebra}{Colombeau algebra}}



\end{itemize}
[[!redirects Colombeau algebras]]



\end{document}