Continuation of Direct Products of Distributions

Abstract

If, in some problems, one has to deal with the ``product'' of distributions ${\mathrm{f}}_{\mathrm{i}}$ (also called generalized functions) $\bar{\mathrm{T}}={\mathrm{\Pi }}_{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{f}}_{\mathrm{i}}$, this product has a priori no definite meaning as a functional $(\bar{\mathrm{T}},\varphi )$ for $\varphi \in \mathrm{S}$. But if ${\mathrm{x}}^{\kappa +1}{\mathrm{\Pi }}_{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{f}}_{\mathrm{i}}$ exists, whatever the associativity is between some powers ${\mathrm{r}}_{\mathrm{i}}$ of $\mathrm{x}$ (${\mathrm{r}}_{\mathrm{i}}\in \mathbb{N},{\sum }_{\mathrm{i}}{\mathrm{r}}_{\mathrm{i}}\le \kappa +1,{\mathrm{r}}_{\mathrm{i}}\ge 0$) and the various ${\mathrm{f}}_{\mathrm{i}}$, then a continuation of the linear functional $\bar{\mathrm{T}}$ from $\mathrm{M}$ onto ${\mathrm{S}}^{(\mathrm{N})}$ for some $\mathrm{N}$ is shown to exist in such a way that ${\mathrm{x}}^{\kappa +1}\bar{\mathrm{T}}$ is defined unambiguously, and $(\bar{\mathrm{T}},\varphi ),\varphi \in \mathrm{S}$, significant, though not unique.

3 pages, latex