Characterizations of algebras of rapidly decreasing generalized functions

TL;DR

This paper extends the classical characterizations of Schwartz space to the broader algebras of rapidly decreasing generalized functions and their regular counterparts, providing new insights into their structure.

Contribution

It introduces characterizations of algebras of rapidly decreasing generalized functions, expanding the classical Schwartz space framework.

Findings

Extended Schwartz space characterizations to generalized functions

Established properties of regular rapidly decreasing generalized functions

Provided a foundation for further analysis of generalized function algebras

Abstract

The well-known characterizations of Schwartz space $\mathcal{S}$ of rapidly decreasing functions is extended to the algebra $\mathcal{G}_{\mathcal{S}}$ of rapidly decreasing generalized functions and to the algebra $\mathcal{G}_{% \mathcal{S}}^{\infty}$ of regular rapidly decreasing generalized functions.