On The Noncommutative Neutrix Product Of Distributions

Abstract

Let f and g be distributions and let g(n) = (g * delta(n))(x), where delta(n)(x) is a certain sequence converging to the Dirac-delta function delta( x). The noncommutative neutrix product f circle g of f and g is defined to be the neutrix limit of the sequence {f g(n)}, provided the limit h exists in the sense that N-lim(n ->infinity) < f (x) g(n)(x), phi(x)> = < h(x), phi( 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(+)(r) lnx(+)circle x(-r-1) lnx(-) and x(-)(-r-1) lnx(-) circle x(+)(r) lnx(+) are proved to exist and are evaluated for r = 1, 2,.... It is consequently seen that these two products are in fact equal. Copyright (c) 2007 Emin Ozcag 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.