Algebras of generalized functions with smooth parameter dependence

TL;DR

This paper demonstrates the isomorphism between spaces of Colombeau generalized functions with smooth and continuous parameter dependence, and explores algebraic properties of the associated ring of generalized numbers.

Contribution

It introduces a unified framework for generalized functions with smooth parameter dependence and studies the algebraic structure of the related ring of generalized numbers.

Findings

Spaces with smooth and continuous parameter dependence are isomorphic.

The algebraic and order structures of the ring of generalized numbers are characterized.

Properties of ideals in the ring are established.

Abstract

We show that spaces of Colombeau generalized functions with smooth parameter dependence are isomorphic to those with continuous parametrization. Based on this result we initiate a systematic study of algebraic properties of the ring $\widetilde{\mathbb{K}}_{sm}$ of generalized numbers in this unified setting. In particular, we investigate the ring and order structure of $\widetilde{\mathbb{K}}_{sm}$ and establish some properties of its ideals.