Unifying order structures for Colombeau algebras

TL;DR

This paper introduces a unified framework for Colombeau algebras using a generalized order structure, simplifying their definitions and enabling broader applications in nonlinear generalized functions.

Contribution

It proposes a general set of indices and a generalized big-O notation to unify various Colombeau-type algebras and streamline their formal definitions.

Findings

Unified presentation of Colombeau algebras

Generalized big-O preserves key properties

Effective application to pointwise characterization

Abstract

We define a general notion of set of indices which, using concepts from pre-ordered sets theory, permits to unify the presentation of several Colombeau-type algebras of nonlinear generalized functions. In every set of indices it is possible to generalize Landau's notion of big-O such that its usual properties continue to hold. Using this generalized notion of big-O, these algebras can be formally defined the same way as the special Colombeau algebra. Finally, we examine the scope of this formalism and show its effectiveness by applying it to the proof of the pointwise characterization in Colombeau algebras.