Brian Fisher, Cheng Lin-Zhi
The product of distributions on $R^m$

Comment.Math.Univ.Carolinae 33,4 (1992) 605-614.

Abstract:The fixed infinitely differentiable function $\rho (x)$ is such that $\{n\rho (n x)\}$ is a regular sequence converging to the Dirac delta function $\delta $. The function $\delta _{\bold n}(\bold x)$, with $\bold x=(x_1,..., x_m)$ is defined by $$ \delta _{\bold n}(\bold x)=n_1 \rho (n_1 x_1)...n_m \rho (n_m x_m). $$ The product $f \circ g$ of two distributions $f$ and $g$ in $\Cal D'_m$ is the distribution $h$ defined by $$ N-\lim -{n_1\rightarrow \infty }...N-\lim -{n_m\rightarrow \infty }\langle f_{\bold n} g_{\bold n}, \phi \rangle = \langle h, \phi \rangle , $$ provided this neutrix limit exists for all $\phi (\bold x)=\phi _1(x_1)...\phi _m(x_m)$, where $f_{\bold n}=f *\delta _{\bold n}$ and $g_{\bold n}=g*\delta _{\bold n}$.

Keywords: distribution, neutrix limit, neutrix product
AMS Subject Classification: 46F10