Colombeau Algebra

In mathematics, the Colombeau algebra (named for Jean-François Colombeau) is an algebra introduced with the aim of constructing an improved theory of distributions in which multiplication is not problematic. The origins of the theory are in applications to quasilinear hyperbolic partial differential equations.

It is defined as a quotient algebra

Here the moderate functions on Rn are defined as

which are families (f_ε) of smooth functions on Rn such that

(where R₊=(0,∞)) is the set of "regularization" indices, and for all compact subsets K of Rn and multiindices α we have N > 0 such that

The ideal of negligible functions is defined in the same way but with the partial derivatives instead bounded by O(εN) for all N > 0.