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.
Contents
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
-
D ′ ( R ) is linearly embedded intoA ( R ) such that the constant function1 becomes the unity inA ( R ) , - There is a partial derivative operator
∂ onA ( R ) which is linear and satisfies the Leibniz rule, - the restriction of
∂ toD ′ ( R ) coincides with the usual partial derivative, - the restriction of
∘ toC ( R ) × C ( R ) coincides with the pointwise product.
However, L. Schwartz' result implies that these requirements cannot hold simultaneously. The same is true even if, in 4., one replaces
Colombeau algebras are constructed to satisfy conditions 1.–3. and a condition like 4., but with
Basic idea
It is defined as a quotient algebra
Here the moderate functions on
which are families (fε) of smooth functions on
(where R+ = (0,∞)) is the set of "regularization" indices, and for all compact subsets K of
The ideal
An introduction to Colombeau Algebras is given in here
Embedding of distributions
The space(s) of Schwartz distributions can be embedded into this simplified algebra by (component-wise) convolution with any element of the algebra having as representative a δ-net, i.e. such that
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 canonic embeddings of distributions. A well known full version is obtained by adding the mollifiers as second indexing set.