An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms

TL;DR

This paper develops an axiomatic differential algebra of generalized functions over a non-Archimedean field, identifies inconsistencies in classical Laplace transform theory, and proposes a contradiction-free alternative within this framework.

Contribution

It introduces a new axiomatic approach to generalized functions, ensuring consistency and addressing issues in traditional Laplace transform theory.

Findings

Established an axiomatic differential algebra containing Schwartz distributions.

Identified an inconsistency in conventional Laplace transform theory.

Proposed a contradiction-free alternative for Laplace transforms within the new algebra.

Abstract

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of 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. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers 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.