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Keywords:
distribution; neutrix limit; neutrix product
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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[4] Fisher B.: The product of the distributions $x_+^{-r-1/2}$ and $x_-^{-r-1/2}$. Proc. Camb. Phil. Soc. 71 (1972), 123-130. MR 0296690 | Zbl 0239.46031
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[6] Fisher B., Li C.K.: On the product of distributions in $m$ variables. Jiangsu Coll. Jnl. 11 (1990), 1-10. MR 1069541
[7] Schwartz L.: Théorie des distributions. Vol. I, II, Herman, 1957. MR 0107812 | Zbl 0962.46025
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