Some Distributional Products of Mikusiński Type in the Colombeau Algebra $\mathcal{G}(R^m)$

Abstract

Particular products of Schwartz distributions on the Euclidean space ${\mathbb{R}}^m$ are derived when the latter have coinciding point singularities and the products are ’balanced’ so that their sum to give an ordinary distribution. These products follow the pattern of a known distributional product published by Jan Mikusiński in 1966. The results are obtained in the Colombeau algebra $\mathcal{G}({\mathbb{R}}^m)$ of generalized functions. $\mathcal{G}({\mathbb{R}}^m)$ is a relevant algebraic construction, with the distribution space linearly embedded, which by the notion of ’association’ allows the results to be evaluated on the level of distributions.