## A commutative neutrix convolution product of distributions.(English)Zbl 0821.46050

Summary: Let $$f$$ and $$g$$ be distributions in $${\mathcal D}'$$ and let $f_ n(x)= f(x) \tau_ n(x),\quad g_ n(x)= g_ n(x) \tau_ n(x),$ where $$\tau_ n(x)$$ is a certain function which converges to the identity function as $$n$$ tends to infinity. Then the neutrix convolution product $$f\boxed{*} g$$ is defined as the neutrix limit of the sequence $$\{f_ n * g_ n\}$$, provided the limit $$h$$ exists in the sense that $N- \lim_{n\to \infty} \langle f_ n * g_ n, \phi\rangle= \langle h, \phi\rangle$ for all $$\phi$$ in $$\mathcal D$$. The neutrix convolution products $$x^ \lambda_ - \boxed{*} x^ n_ +$$ for $$\lambda, \mu,\lambda+ \mu\neq 0$$, $$\pm 1,\pm 2,\dots$$ and $$x^ \lambda_ -\boxed{*} x^ s_ +$$ for $$\lambda\neq 0$$, $$\pm 1, \pm 2,\dots$$ and $$s= 0, 1, 2,\dots$$ are evaluated, from which other neutrix convolution products are deduced.

### MSC:

 46F10 Operations with distributions and generalized functions

### Keywords:

neutrix convolution product