Algebraic Approach to Colombeau Theory

TL;DR

This paper develops a differential algebra of generalized functions extending Schwartz distributions, based on axioms in algebra and topology, aiming to make Colombeau theory more accessible to mathematicians and scientists outside the specialized community.

Contribution

It introduces a new axiomatic differential algebra of generalized functions of Colombeau type with a focus on algebraic and topological foundations, and discusses its uniqueness and consistency.

Findings

Contains a copy of Schwartz distributions

Regular distributions with smooth kernels form a subalgebra

Ensures axioms are consistent and independent

Abstract

We present a differential algebra of generalized functions over a field of generalized scalars by means of several axioms in terms of general algebra and topology. Our differential algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions, and the set of regular distributions with $\mathcal C^\infty$-kernels forms a differential subalgebra. We discuss the uniqueness of the field of scalars as well as the consistency and independence of our axioms. This article is written mostly to satisfy the interest of mathematicians and scientists who do not necessarily belong to the \emph{Colombeau community}; that is to say, those who do not necessarily work in the \emph{non-linear theory of generalized functions}.