The fixed infinitely differentiable function $\rho (x)$ is such that $\{n\rho (n x)\}$ is a re\-gular sequence converging to the Dirac delta function $\delta $. The function $\delta _{\bold n}(\bold x)$, with $\bold x=(x_1, \dots , x_m)$ is defined by $$ \delta _{\bold n}(\bold x)=n_1 \rho (n_1 x_1)\dots n_m \rho (n_m x_m). $$ The product $f \circ g$ of two distributions $f$ and $g$ in $\mathcal D'_m$ is the distribution $h$ defined by $$ \operatornamewithlimits{N\mbox{--}\lim}\limits _{n_1\rightarrow \infty } \dots \operatornamewithlimits{N\mbox{--}\lim}\limits _{n_m\rightarrow \infty } \langle f_{\bold n} g_{\bold n}, \phi \rangle = \langle h, \phi \rangle, $$ provided this neutrix limit exists for all $\phi (\bold x)=\phi _1(x_1)\dots \phi _m(x_m)$, where $f_{\bold n}=f \ast \delta _{\bold n}$ and $g_{\bold n}=g\ast \delta _{\bold n}$.
The fixed infinitely differentiable function $\rho (x)$ is such that $\{n\rho (n x)\}$ is a re\-gular sequence converging to the Dirac delta function $\delta $. The function $\delta _{\bold n}(\bold x)$, with $\bold x=(x_1, \dots , x_m)$ is defined by $$ \delta _{\bold n}(\bold x)=n_1 \rho (n_1 x_1)\dots n_m \rho (n_m x_m). $$ The product $f \circ g$ of two distributions $f$ and $g$ in $\mathcal D'_m$ is the distribution $h$ defined by $$ \operatornamewithlimits{N\mbox{--}\lim}\limits _{n_1\rightarrow \infty } \dots \operatornamewithlimits{N\mbox{--}\lim}\limits _{n_m\rightarrow \infty } \langle f_{\bold n} g_{\bold n}, \phi \rangle = \langle h, \phi \rangle, $$ provided this neutrix limit exists for all $\phi (\bold x)=\phi _1(x_1)\dots \phi _m(x_m)$, where $f_{\bold n}=f \ast \delta _{\bold n}$ and $g_{\bold n}=g\ast \delta _{\bold n}$.
@article{CMUC_1992_33_4_a4,
author = {Lin-Zhi, Cheng and Fisher, Brian},
title = {The product of distributions on $R^m$},
journal = {Commentationes Mathematicae Universitatis Carolinae},
pages = {605--614},
year = {1992},
volume = {33},
number = {4},
mrnumber = {1240181},
zbl = {0818.46035},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMUC_1992_33_4_a4/}
}
TY - JOUR
AU - Lin-Zhi, Cheng
AU - Fisher, Brian
TI - The product of distributions on $R^m$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1992
SP - 605
EP - 614
VL - 33
IS - 4
UR - http://geodesic.mathdoc.fr/item/CMUC_1992_33_4_a4/
LA - en
ID - CMUC_1992_33_4_a4
ER -
%0 Journal Article
%A Lin-Zhi, Cheng
%A Fisher, Brian
%T The product of distributions on $R^m$
%J Commentationes Mathematicae Universitatis Carolinae
%D 1992
%P 605-614
%V 33
%N 4
%U http://geodesic.mathdoc.fr/item/CMUC_1992_33_4_a4/
%G en
%F CMUC_1992_33_4_a4
Lin-Zhi, Cheng; Fisher, Brian. The product of distributions on $R^m$. Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992) no. 4, pp. 605-614. http://geodesic.mathdoc.fr/item/CMUC_1992_33_4_a4/
