# Article

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Keywords:
distribution; neutrix limit; neutrix product
Summary:
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 $g_{\bold n}=g\ast \delta _{\bold n}$.
References:
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