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### The product of distributions on ${R}^{m}$

Commentationes Mathematicae Universitatis Carolinae

The fixed infinitely differentiable function $\rho \left(x\right)$ is such that $\left\{n\rho \left(nx\right)\right\}$ is a regular sequence converging to the Dirac delta function $\delta$. The function ${\delta }_{𝐧}\left(𝐱\right)$, with $𝐱=\left({x}_{1},\cdots ,{x}_{m}\right)$ is defined by ${\delta }_{𝐧}\left(𝐱\right)={n}_{1}\rho \left({n}_{1}{x}_{1}\right)\cdots {n}_{m}\rho \left({n}_{m}{x}_{m}\right).$ The product $f\circ g$ of two distributions $f$ and $g$ in ${𝒟}_{m}^{\text{'}}$ is the distribution $h$ defined by $\underset{{n}_{1}\to \infty }{error}\cdots \underset{{n}_{m}\to \infty }{error}〈{f}_{𝐧}{g}_{𝐧},\phi 〉=〈h,\phi 〉,$ provided this neutrix limit exists for all $\phi \left(𝐱\right)={\phi }_{1}\left({x}_{1}\right)\cdots {\phi }_{m}\left({x}_{m}\right)$, where ${f}_{𝐧}=f*{\delta }_{𝐧}$ and ${g}_{𝐧}=g*{\delta }_{𝐧}$.

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