Editing Colombeau algebra (section)

== Basic idea ==

The Colombeau Algebra<ref>{{cite arXiv|last = Gratus|first =  J.|title = Colombeau Algebra: A pedagogical introduction|year = 2013|class = math.FA|eprint=1308.0257}}</ref> is defined as the [[quotient associative algebra|quotient algebra]]

:<math>C^\infty_M(\mathbb{R}^n)/C^\infty_N(\mathbb{R}^n).</math>

Here the algebra of ''moderate functions'' <math>C^\infty_M(\mathbb{R}^n)</math> on <math>\mathbb{R}^n</math> is the algebra of families of smooth ''regularisations'' (''f<sub>ε</sub>'')

:<math>{f:} \mathbb{R}_+ \to C^\infty(\mathbb{R}^n)</math>
of [[smooth function]]s on <math>\mathbb{R}^n</math>
(where '''R'''<sub>+</sub>&nbsp;=&nbsp;(0,∞) is the "[[regularization (mathematics)|regularization]]" parameter ε), such that for all compact subsets ''K'' of <math>\mathbb{R}^n</math> and all [[multiindices]] α, there is an ''N'' > 0 such that

:<math>\sup_{x\in K}\left|\frac{\partial^{|\alpha|}}{(\partial x_1)^{\alpha_1}\cdots(\partial x_n)^{\alpha_n}}f_\varepsilon(x)\right| = O(\varepsilon^{-N})\qquad(\varepsilon\to 0).</math>

The [[ideal (ring theory)|ideal]] <math>C^\infty_N(\mathbb{R}^n)</math> of ''negligible functions'' is defined in the same way but with the partial derivatives instead bounded by O(''ε<sup>+N</sup>'') for '''all''' ''N'' > 0.