Editing Colombeau algebra

In [[mathematics]], a '''Colombeau algebra''' is an [[associative algebra|algebra]] of a certain kind containing the space of [[distribution (mathematics)|Schwartz distributions]]. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras provide a rigorous framework for this.

Such a multiplication of distributions has long been believed to be impossible because of L. Schwartz' impossibility result, which basically states that there cannot be a differential algebra containing the space of distributions and preserving the product of continuous functions. However, if one only wants to preserve the product of smooth functions instead such a construction becomes possible, as demonstrated first by Colombeau.

As a mathematical tool, Colombeau algebras can be said to combine a treatment of singularities, differentiation and nonlinear operations in one framework, lifting the limitations of distribution theory. These algebras have found numerous applications in the fields of partial differential equations, geophysics, [[microlocal analysis]] and [[general relativity]] so far {{Dubious|date=August 2024}}.

Colombeau algebras are named after French mathematician [[Jean François Colombeau]].

== Schwartz' impossibility result ==

Attempting to embed the space <math>\mathcal{D}'(\mathbb{R})</math> of distributions on <math>\mathbb{R}</math> into an associative algebra <math>(A(\mathbb{R}), \circ, +)</math>, the following requirements seem to be natural:

# <math>\mathcal{D}'(\mathbb{R})</math> is linearly embedded into <math>A(\mathbb{R})</math> such that the constant function <math>1</math> becomes the unity in <math>A(\mathbb{R})</math>,
# There is a [[partial derivative]] operator <math>\partial</math> on <math>A(\mathbb{R})</math> which is linear and satisfies the Leibniz rule,
# the restriction of <math>\partial</math> to <math>\mathcal{D}'(\mathbb{R})</math> coincides with the usual partial derivative,
# the restriction of <math>\circ</math> to <math>C(\mathbb{R}) \times C(\mathbb{R})</math> coincides with the pointwise product.

However, L. Schwartz' result<ref>L. Schwartz, 1954, "Sur l'impossibilité de la multiplication des distributions", ''Comptes Rendus de L'Académie des Sciences'' 239, pp. 847–848 [http://gallica.bnf.fr/ark:/12148/bpt6k3191m/f847.image.langFR]</ref> implies that these requirements cannot hold simultaneously. The same is true even if, in 4., one replaces <math>C(\mathbb{R})</math> by <math>C^k(\mathbb{R})</math>, the space of <math>k</math> times continuously differentiable functions. While this result has often been interpreted as saying that a general multiplication of distributions is not possible, in fact it only states that one cannot unrestrictedly combine differentiation, multiplication of continuous functions and the presence of singular objects like the Dirac delta.

Colombeau algebras are constructed to satisfy conditions 1.–3. and a condition like 4., but with <math>C(\mathbb{R}) \times C(\mathbb{R})</math> replaced by <math>C^\infty(\mathbb{R}) \times C^\infty(\mathbb{R})</math>, i.e., they preserve the product of smooth (infinitely differentiable) functions only.

== 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.

== Embedding of distributions ==
The space(s) of [[Schwartz distribution]]s can be embedded into the ''simplified'' algebra by (component-wise) [[convolution]] with any element of the algebra having as representative a ''[[e-net (probability theory)|&delta;-net]]'', i.e. a family of smooth functions <math>\varphi_\varepsilon</math> such that <math>\varphi_\varepsilon\to\delta</math> in '' D' '' as&nbsp;''ε''&nbsp;→&nbsp;0.

This embedding is non-canonical, because it depends on the choice of the δ-net. However, there are versions of Colombeau algebras (so called ''full'' algebras) which allow for canonical embeddings of distributions. A well known ''full'' version is obtained by adding the [[Mollifier|mollifiers]] as second indexing set.

== See also ==
* [[Generalized function]]

== Notes ==
{{Reflist}}

== References ==
* Colombeau, J. F., ''New Generalized Functions and Multiplication of the Distributions''. North Holland, Amsterdam, 1984.
* Colombeau, J. F., ''Elementary introduction to new generalized functions''. North-Holland, Amsterdam, 1985.
* Nedeljkov, M., [[Stevan Pilipović|Pilipović, S.]], Scarpalezos, D., ''Linear Theory of Colombeau's Generalized Functions'', Addison Wesley, Longman, 1998.
* Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.; ''Geometric Theory of Generalized Functions with Applications to General Relativity'', Springer Series Mathematics and Its Applications, Vol. 537, 2002; {{isbn|978-1-4020-0145-1}}.

[[Category:Smooth functions]]
[[Category:Functional analysis]]
[[Category:Algebras]]
[[Category:Schwartz distributions]]