On the characterization of p-adic Colombeau-Egorov generalized functions by their point values

Abstract

We show that contrary to recent papers by S. Albeverio, A. Yu. Khrennikov and V. Shelkovich, point values do not determine elements of the so-called p-adic Colombeau-Egorov algebra uniquely. We further show in a more general way that for an Egorov algebra of generalized functions on a locally compact ultrametric space (M,d) taking values in a non-trivial ring, a point value characterization holds if and only if (M,d) is discrete. Finally, following an idea due to M. Kunzinger and M. Oberguggenberger, a generalized point value characterization of such an Egorov algebra is given. Elements of the latter are constructed which differ from the p-adic delta-distribution considered as a generalized function, yet coincide on point values with the latter.