Asymptotic gauges: generalization of Colombeau type algebras

TL;DR

This paper introduces a unified framework for Colombeau-type algebras of generalized functions using asymptotic gauges, enabling broader applications including solving linear ODEs with generalized coefficients.

Contribution

It generalizes Colombeau algebras via asymptotic gauges, encompassing special, full, and nonstandard types, and establishes existence and uniqueness results for solving linear ODEs within this framework.

Findings

Unified framework for Colombeau algebras using asymptotic gauges

Existence of minimal algebra for solving linear ODEs with generalized coefficients

Broader applicability of Colombeau algebras compared to traditional special algebra

Abstract

We use the general notion of set of indices to construct algebras of nonlinear generalized functions of Colombeau type. They are formally defined in the same way as the special Colombeau algebra, but based on more general growth condition formalized by the notion of asymptotic gauge. This generalization includes the special, full and nonstandard analysis based Colombeau type algebras in a unique framework. We compare Colombeau algebras generated by asymptotic gauges with other analogous construction, and we study systematically their properties, with particular attention to the existence and definition of embeddings of distributions. We finally prove that, in our framework, for every linear homogeneous ODE with generalized coefficients there exists a minimal Colombeau algebra generated by asymptotic gauges in which the ODE can be uniquely solved. This marks a main difference with the Colombeau special algebra, where only linear homogeneous ODEs satisfying some restriction on the coefficients can be solved.