The product of distributions on $R^m$
Lin-Zhi, Cheng ; Fisher, Brian
Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992), p. 605-614 / Harvested from Czech Digital Mathematics Library
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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 $g_{\bold n}=g\ast \delta _{\bold n}$.

Publié le : 1992-01-01
Classification:  46F10 
@article{118531,
     author = {Cheng Lin-Zhi and Brian Fisher},
     title = {The product of distributions on $R^m$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {33},
     year = {1992},
     pages = {605-614},
     zbl = {0818.46035},
     mrnumber = {1240181},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118531}
}

Lin-Zhi, Cheng; Fisher, Brian. The product of distributions on $R^m$. Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992) pp. 605-614. http://gdmltest.u-ga.fr/item/118531/

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