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 $f\circ g$ of $f$ and $g$ is defined to be the neutrix limit of the sequence $\left\{f{g}_n\right\}$, provided the limit $h$ exists in the sense that $\text{N‐}{lim}_{n\to \infty }\langle f\left(x\right)g_n\left(x\right),\phi \left(x\right)\rangle =\langle h\left(x\right),\phi \left(x\right)\rangle$, 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 $x_{+}^{r}ln{x}_{+}\circ x_{-}^{-r-1}ln{x}_{-}$ and $x_{-}^{-r-1}ln{x}_{-}\circ x_{+}^{r}ln{x}_{+}$ are proved to exist and are evaluated for $r=1,2,\dots$. It is consequently seen that these two products are in fact equal.