On association in Colombeau algebras without asymptotics

TL;DR

This paper explores how the concept of association with distributions can be adapted to a variant of Colombeau algebras that does not rely on asymptotic estimates, highlighting the ongoing importance of asymptotics.

Contribution

It demonstrates that association in non-asymptotic Colombeau algebras still fundamentally depends on asymptotic concepts, clarifying their role in generalized function theory.

Findings

Association can be transferred to non-asymptotic Colombeau algebras

Asymptotics remain essential for defining association

Clarifies the role of asymptotics in generalized functions

Abstract

A recent variant of Colombeau algebras does not employ asymptotic estimates for its definition. We discuss how the concept of association with distributions transfers to this setting and why it still needs to be based on asymptotics.