The product of distributions on \(R^ m\). (English) Zbl 0818.46035

Fisher, Brian; Cheng, Linzhi

Commentat. Math. Univ. Carol. 33, No. 4, 605-614 (1992).

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