On balanced products of the distributions $x_{\pm }^a$ in Colombeau algebra $\EuScript{G}(\mathbb{R})$

Abstract

Results on products of the distributions $x_{\pm }^a$ and $\de^{(p)}(x)$ with coinciding singularities are derived when the products are balanced so that they exist in the distribution space. These results follow the idea of a known result on distributional products published by Jan Mikusiński in 1966. They are obtained in Colombeau algebra $\EuScript{G}(\mathbb{R})$ of generalized functions that contains the distribution space and the notion of 'association' in it allows obtaining results in terms of distributions.

14 pages