Continuation of Direct Products of Distributions

Abstract

If, in some problems, one has to deal with the ``product'' of distributions $\rm f_i$ (also called generalized functions) $\rm\bar T = Π^m_{i=1} f_i$, this product has a priori no definite meaning as a functional $(\rm \bar T, ϕ) $ for $\rmϕ\in S$. But if $\rm x^{κ+1} Π^m_{i=1} f_i$ exists, whatever the associativity is between some powers $\rm r_i$ of $\rm x$ ($\rm r_i \in \Bbb N, \sum_i r_i\leq κ+1, r_i \geq 0$) and the various $\rm f_i$, then a continuation of the linear functional $\rm \bar T$ from $\rm M$ onto $\rm S^{(N)}$ for some $\rm N$ is shown to exist in such a way that $\rm x^{κ+1} \bar T$ is defined unambiguously, and $\rm (\bar T, ϕ), ϕ\in S$, significant, though not unique.