On positivity in algebras of tempered generalized functions

TL;DR

This paper demonstrates that in algebras of tempered generalized functions, positivity and invertibility cannot be characterized pointwise, contrasting with the Colombeau algebra, and provides a refined point value characterization.

Contribution

It provides a counterexample showing the failure of pointwise positivity and invertibility characterization in tempered generalized function algebras and refines the point value characterization.

Findings

Positivity and invertibility cannot be characterized pointwise in tempered generalized function algebras.

A counterexample illustrating this failure is constructed.

The point value characterization of tempered generalized functions is refined.

Abstract

An explicit counterexample shows that contrary to the situation in the special Colombeau algebra, positivity and invertibility cannot be characterized pointwise in algebras of tempered generalized functions. Further a point value characterization of the latter is refined.