Topology and regularity for generalized ultradistribution algebras

TL;DR

This paper investigates the properties of generalized ultradistribution algebras, establishing conditions for strong association, negligibility, and translation invariance, thereby advancing the understanding of their structure and regularity.

Contribution

It provides new results on strong association, negligibility, and translation invariance in generalized ultradistribution algebras, linking regularity conditions to ultradifferentiable functions.

Findings

Strong association implies ultradifferentiability under regularity conditions.

Weakly negligible nets are negligible in the generalized ultradistribution sense.

Translation invariant ultradistributions are constants in the algebra.

Abstract

Compiling essential results for non-quasianalytic ultradistribution spaces and Colombeau versions of generalized ultradistribution algebras, we analyze strong $B$- and strong $R$-association of a generalized ultradistribution $[(f_\varepsilon)]$. The strong association of $[(f_\varepsilon)]$ to a Komatsu-type ultradistribution $T$, with additional assumption on regularity of $[(f_\varepsilon)]$ of Beurling, respectively, Roumieu type, implies that $T$ is an ultradifferentiable function of Beurling, Roumieu type, respectively. We demonstrate that, under suitable conditions on regularity, a weakly negligible net $(g_\varepsilon)_{\varepsilon\in(0,1)}$ (meaning that the net of complex numbers $(\int g_\varepsilon\phi dx)_{\varepsilon\in(0,1)}$ is Beurling, respectively, Roumieu negligible for every ultradifferentiable function $\phi$ in the corresponding test space), is a negligible net in the sense of generalized ultradistributions. Furthermore, we prove that a translation invariant generalized ultradistribution $g$ is equal to a generalized constant in both types of generalized ultradistribution algebras.