Continuation of Direct Products of Distributions

TL;DR

This paper investigates the extension of products of distributions, establishing conditions under which a linear functional involving these products can be unambiguously continued from a space of test functions to a larger space, despite inherent ambiguities.

Contribution

It provides a method to extend the product of distributions as a linear functional, ensuring well-definedness under specific conditions involving powers of x.

Findings

Existence of continuation of the linear functional for certain products of distributions.

Unambiguous definition of x^{k+1} times the distribution under specified conditions.

The continuation is significant but not unique.

Abstract

If, in some problems, one has to deal with the ``product'' of distributions $\rm f_i$ (also called generalized functions) $\rm\bar T = \Pi^m_{i=1} f_i$, this product has a priori no definite meaning as a functional $(\rm \bar T, \phi) $ for $\rm\phi \in S$. But if $\rm x^{\kappa +1} \Pi^m_{i=1} f_i$ exists, whatever the associativity is between some powers $\rm r_i$ of $\rm x$ ($\rm r_i \in \Bbb N, \sum_i r_i\leq \kappa +1, r_i \geq 0$) 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 $\rm x^{\kappa +1} \bar T$ is defined unambiguously, and $\rm (\bar T, \phi), \phi \in S$, significant, though not unique.