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 ${\delta}_n(x)$. The noncommutative neutrix product $f \circ g$ 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 $\textrm{N-}\lim_{n\to \infty } \langle f(x) g_n(x), \phi (x) \rangle = \langle h(x) , \phi (x) \rangle $, for all test functions in $\mathfrak{D}$. In this paper, using the concept of the neutrix limit due to van der Corput (1960), the noncommutative neutrix products $ x_+^r \ln x_+ \circ x_-^{-r-1} \ln x_- $ and $x_-^{-r-1} \ln x_- \circ x_+^r \ln x_+$ are proved to exist and are evaluated for $r=1, 2, \ldots $. It is consequently seen that these two products are in fact equal.