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On the characterization of p ‐adic Colombeau–Egorov generalized functions by their point values
Author(s) -
Mayerhofer Eberhard
Publication year - 2007
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200510554
Subject(s) - mathematics , characterization (materials science) , ultrametric space , point (geometry) , element (criminal law) , space (punctuation) , pure mathematics , distribution (mathematics) , mathematical analysis , metric space , geometry , physics , linguistics , philosophy , political science , law , optics
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)