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

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 (ℚ np ) uniquely. We further show in a more general way that for an Egorov algebra ( M, R ) of generalized functions on a locally compact ultrametric space ( M, d ) taking values in a nontrivial 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 ( M, R ) is given. Elements of (ℚ np ) are constructed which differ from the p ‐adic δ ‐distribution considered as an element of (ℚ np ), yet coincide on point values with the latter. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)