Multiplication of distributions in any dimension: applications to $\delta$-function and its derivatives

TL;DR

This paper introduces a new method for multiplying distributions, especially delta functions and their derivatives, applicable in any dimension, motivated by engineering applications and extending previous one-dimensional approaches.

Contribution

A novel, easily extendable definition of distribution multiplication that allows for multiplication of delta functions and derivatives in any dimension.

Findings

Delta functions can be multiplied using the new definition.

The multiplication method extends to any spatial dimension.

Application potential in engineering analysis.

Abstract

In two previous papers the author introduced a multiplication of distributions in one dimension and he proved that two one-dimensional Dirac delta functions and their derivatives can be multiplied, at least under certain conditions. Here, mainly motivated by some engineering applications in the analysis of the structures, we propose a different definition of multiplication of distributions which can be easily extended to any spatial dimension. In particular we prove that with this new definition delta functions and their derivatives can still be multiplied.