The functional analytic foundation of Colombeau algebras

TL;DR

This paper introduces a unifying functional analytic framework for Colombeau algebras, clarifying their structure and interrelations, which enhances understanding and potential applications in nonlinear distribution operations.

Contribution

It provides a comprehensive functional analytic foundation that unifies various Colombeau algebra variants, clarifying their structural properties and relationships.

Findings

Unified hierarchy of Colombeau algebras established

Structural properties of different variants clarified

Relations between variants made explicit

Abstract

Colombeau algebras constitute a convenient framework for performing nonlinear operations like multiplication on Schwartz distributions. Many variants and modifications of these algebras exist for various applications. We present a functional analytic approach placing these algebras in a unifying hierarchy, which clarifies their structural properties as well as their relation to each other.