Colombeau algebra

In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of 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.
Schwartz' impossibility result.
Attempting to embed the space formula_1 of distributions on formula_2 into an associative algebra formula_3, the following requirements seem to be natural:
However, L. Schwartz' result implies that these requirements cannot hold simultaneously. The same is true even if, in 4., one replaces formula_14 by formula_15, the space of formula_16 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 formula_13 replaced by formula_18, i.e., they preserve the product of smooth (infinitely differentiable) functions only.
Basic idea.
The Colombeau Algebra is defined as the quotient algebra
Here the algebra of "moderate functions" formula_20 on formula_21 is the algebra of families of smooth "regularisations" ("fε")
of smooth functions on formula_21
(where R+ = (0,∞) is the "regularization" parameter ε), such that for all compact subsets "K" of formula_21 and all multiindices α, there is an "N" > 0 such that
The ideal formula_26 of "negligible functions" is defined in the same way but with the partial derivatives instead bounded by O("ε+N") for all "N" > 0.
Embedding of distributions.
The space(s) of Schwartz distributions can be embedded into the "simplified" algebra by (component-wise) convolution with any element of the algebra having as representative a "δ-net", i.e. a family of smooth functions formula_27 such that formula_28 in " D' " as "ε" → 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 mollifiers as second indexing set.