A new approach to temperate generalized functions

TL;DR

This paper introduces a novel construction of the algebra G_{ au} of temperate nonlinear generalized functions using the space O_{M}, enhancing its properties and integrating a regularity theory and Fourier transform within this framework.

Contribution

It presents a new approach to G_{ au} based on O_{M}, improving its structure and compatibility with Colombeau algebras, and introduces a regularity theory and Fourier transform for these functions.

Findings

G_{ au} constructed on O_{M} with better properties.

Introduction of a regularity theory within G_{ au}.

Definition of G_{O_{C prime}} as the Fourier image of G_{ au}.

Abstract

A new approach to the algebra G_{\tau} of temperate nonlinear generalized functions is proposed, in which G_{\tau} is based on the space O_{M} endowed with is natural topology in contrary to previous constructions. Thus, this construction fits perfectly in the general scheme of construction of Colombeau type algebras and reveals better properties of G_{\tau}. This is illustrated by the natural introduction of a regularity theory in G_{\tau}, of the Fourier transform, with the definition of G_{O_{C prime}}, the space of rapidly generalized distributions which is the Fourier image of G_{\tau}.