On the Noncommutative Neutrix Product of Distributions

Let fand 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 . 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