Colombeau Algebra - Embedding of Distributions

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 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 canonic embeddings of distributions. A well known full version is obtained by adding the mollifiers as second indexing set.