Topology and regularity for generalized ultradistribution algebras

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 $[(f_{\epsilon })]$. The strong association of $[(f_{\epsilon })]$ to a Komatsu-type ultradistribution $T$, with additional assumption on regularity of $[(f_{\epsilon })]$ 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 $(g_{\epsilon }{)}_{\epsilon \in (0,1)}$ (meaning that the net of complex numbers $(\int g_{\epsilon }\phi dx{)}_{\epsilon \in (0,1)}$ 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.