Abstract and Applied Analysis
Volume 2007 (2007), Article ID 81907, 10 pages
doi:10.1155/2007/81907
Research Article

On the Noncommutative Neutrix Product of Distributions

Emin Özçaḡ,1 İnci Ege,1 Haşmet Gürçay,1 and Biljana Jolevska-Tuneska2

1Department of Mathematics, University of Hacettepe, Ankara 06532, Beytepe, Turkey
2Faculty of Electrical Engineering, Karpos II bb, 9100, Skopje, Macedonia

Received 21 August 2007; Accepted 5 November 2007

Academic Editor: Agacik Zafer

Copyright © 2007 Emin Özçaḡ et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let f and g be distributions and let gn=(g*δn)(x), where δn(x) is a certain sequence converging to the Dirac-delta function δ(x). The noncommutative neutrix product f∘g of f and g is defined to be the neutrix limit of the sequence {fgn}, provided the limit h exists in the sense that N‐limn→∞〈f(x)gn(x),φ(x)〉=〈h(x),φ(x)〉, for all test functions in 𝒟. In this paper, using the concept of the neutrix limit due to van der Corput (1960), the noncommutative neutrix products x+rlnx+∘x−−r−1lnx− and x−−r−1lnx−∘x+rlnx+ are proved to exist and are evaluated for r=1,2,…. It is consequently seen that these two products are in fact equal.