Editing Colombeau algebra (section)

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]].