Internal sets and internal functions in Colombeau theory

TL;DR

This paper explores internal sets and functions within Colombeau algebras, inspired by nonstandard analysis, establishing a saturation principle and demonstrating applications to the structure of generalized functions.

Contribution

It introduces the concepts of internal sets and functions in Colombeau theory and proves a saturation principle, advancing the understanding of their algebraic and analytical properties.

Findings

Established a saturation principle for internal sets in Colombeau algebras

Defined and analyzed internal functions within the Colombeau framework

Applied internal set theory to derive new results in Colombeau algebra structure

Abstract

Inspired by nonstandard analysis, we define and study internal subsets and internal functions in algebras of Colombeau generalized functions. We prove a saturation principle for internal sets and provide applications to Colombeau algebras.