An axiomatic approach to non-linear theory of generalized functions

TL;DR

This paper develops an axiomatic framework for a Colombeau-type algebra of generalized functions over non-Archimedean fields, enabling nonlinear analysis of distributions with a focus on mathematical consistency.

Contribution

It introduces a new axiomatic approach to generalized functions in non-Archimedean settings, expanding the theoretical foundation beyond classical Colombeau algebras.

Findings

Established axioms for the algebra of generalized functions

Proved uniqueness and consistency of the axiomatic system

Demonstrated the algebra contains Schwartz distributions

Abstract

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of {\em Colombeau type} in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. The article is aimed at mathematicians and physicists who are interested in the non-linear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theory of distributions.