Full Algebra of Generalized Functions and Non-Standard Asymptotic Analysis

TL;DR

This paper develops a new algebra of generalized functions with a canonical embedding of Schwartz distributions, offering a unique multiplication approach, an algebraically closed scalar field, and a connection to non-standard analysis, advancing the mathematical framework for generalized functions.

Contribution

It introduces a novel algebra of generalized functions with a scalar field that is algebraically closed and connects Colombeau theory with non-standard analysis, addressing longstanding issues.

Findings

Scalar set is an algebraically closed field.

Provides a solution to multiplication of Schwartz distributions.

Establishes a link between Colombeau theory and non-standard analysis.

Abstract

We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions. We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau's solution. We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn-Banach extension principle which does not hold in Colombeau theory. We establish a connection between our theory with non-standard analysis and thus answer, although indirectly, a question raised by J.F. Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis.