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The product of distributions on $$R^ m$$. (English) Zbl 0818.46035
Summary: The fixed infinitely differentiable function $$\rho(x)$$ is such that $$\{n\rho (nx)\}$$ 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 $\delta_{{\mathbf n}}({\mathbf x})= n_ 1 \rho(n_ 1 x_ 1)\cdots n_ m \rho(n_ m x_ m).$ The product $$f\circ g$$ of two distributions $$f$$ and $$g$$ in $${\mathcal D}_ m'$$ is the distribution $$h$$ defined by $\text{N}-\lim_{n_ 1\to \infty}\cdots \text{N}-\lim_{n_ m\to \infty} \langle f_{{\mathbf n}} g_{{\mathbf n}}, \phi\rangle= \langle h,\phi\rangle,$ provided this neutrix limit exists for all $$\phi({\mathbf x})= \phi_ 1(x_ 1)\cdots \phi_ m(x_ m)$$, where $$f_{{\mathbf n}}= f * \delta_{{\mathbf n}}$$ and $$g_{{\mathbf n}}= g* \delta_{{\mathbf n}}$$.

##### MSC:
 46F10 Operations with distributions and generalized functions
##### Keywords:
product of distributions; Dirac delta function
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