The product of distributions on $R^m$

Lin-Zhi, Cheng, and Fisher, Brian. "The product of distributions on $R^m$." Commentationes Mathematicae Universitatis Carolinae 33.4 (1992): 605-614. <http://eudml.org/doc/247358>.

@article{Lin1992,
abstract = {The fixed infinitely differentiable function $\rho (x)$ is such that $\lbrace n\rho (n x)\rbrace $ is a regular sequence converging to the Dirac delta function $\delta $. The function $\delta _\{\mathbf \{n\}\}(\mathbf \{x\})$, with $\mathbf \{x\}=(x_1, \dots , x_m)$ is defined by \[ \delta \_\{\mathbf \{n\}\}(\mathbf \{x\})=n\_1 \rho (n\_1 x\_1)\dots n\_m \rho (n\_m x\_m). \] The product $f \circ g$ of two distributions $f$ and $g$ in $\mathcal \{D\}^\{\prime \}_m$ is the distribution $h$ defined by \[ \operatornamewithlimits\{N\mbox\{--\}\lim \}\limits \_\{n\_1\rightarrow \infty \} \dots \operatornamewithlimits\{N\mbox\{--\}\lim \}\limits \_\{n\_m\rightarrow \infty \} \langle f\_\{\mathbf \{n\}\} g\_\{\mathbf \{n\}\}, \phi \rangle = \langle h, \phi \rangle , \] provided this neutrix limit exists for all $\phi (\mathbf \{x\})=\phi _1(x_1)\dots \phi _m(x_m)$, where $f_\{\mathbf \{n\}\}=f \ast \delta _\{\mathbf \{n\}\}$ and $g_\{\mathbf \{n\}\}=g\ast \delta _\{\mathbf \{n\}\}$.},
author = {Lin-Zhi, Cheng, Fisher, Brian},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {distribution; neutrix limit; neutrix product; product of distributions; Dirac delta function},
language = {eng},
number = {4},
pages = {605-614},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The product of distributions on $R^m$},
url = {http://eudml.org/doc/247358},
volume = {33},
year = {1992},
}

TY - JOUR
AU - Lin-Zhi, Cheng
AU - Fisher, Brian
TI - The product of distributions on $R^m$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1992
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 33
IS - 4
SP - 605
EP - 614
AB - The fixed infinitely differentiable function $\rho (x)$ is such that $\lbrace n\rho (n x)\rbrace $ is a regular sequence converging to the Dirac delta function $\delta $. The function $\delta _{\mathbf {n}}(\mathbf {x})$, with $\mathbf {x}=(x_1, \dots , x_m)$ is defined by \[ \delta _{\mathbf {n}}(\mathbf {x})=n_1 \rho (n_1 x_1)\dots n_m \rho (n_m x_m). \] The product $f \circ g$ of two distributions $f$ and $g$ in $\mathcal {D}^{\prime }_m$ is the distribution $h$ defined by \[ \operatornamewithlimits{N\mbox{--}\lim }\limits _{n_1\rightarrow \infty } \dots \operatornamewithlimits{N\mbox{--}\lim }\limits _{n_m\rightarrow \infty } \langle f_{\mathbf {n}} g_{\mathbf {n}}, \phi \rangle = \langle h, \phi \rangle , \] provided this neutrix limit exists for all $\phi (\mathbf {x})=\phi _1(x_1)\dots \phi _m(x_m)$, where $f_{\mathbf {n}}=f \ast \delta _{\mathbf {n}}$ and $g_{\mathbf {n}}=g\ast \delta _{\mathbf {n}}$.
LA - eng
KW - distribution; neutrix limit; neutrix product; product of distributions; Dirac delta function
UR - http://eudml.org/doc/247358
ER -