Weak homogeneity in generalized function algebras

TL;DR

This paper characterizes weakly homogeneous generalized functions within Colombeau algebras, establishing their equivalence to generalized distributions and providing multiple criteria for equality in this context.

Contribution

It offers new characterizations of weak homogeneity and equality in Colombeau algebras, linking them closely to distribution theory.

Findings

Weakly homogeneous generalized functions are characterized up to generalized distribution equality.

Several criteria for equality in the sense of generalized distributions are established.

Results resemble classical distribution theory, bridging it with Colombeau algebra concepts.

Abstract

In this paper, weakly homogeneous generalized functions in the special Colombeau algebras are determined up to equality in the sense of generalized distributions. This yields characterizations that are formally similar to distribution theory. Further, we give several characterizations of equality in the sense of generalized distributions in these algebras.