Some distributional products of Mikusiński type in the Colombeau algebra \({\mathcal G}(R^m)\) (Q5949168)

The famous Mikusinski's result for product of distributions: \(x^{-1}\cdot x^{-1}-\pi^2 \delta(x) \cdot\delta(x) =x^{-2}\), \(x\in \mathbb{R}\) gave a nice idea to the author of this paper to use the Colombeau algebra \(G(R^m)\) in which the space of distributions is embedded \((u\in D'\to \widetilde u\in G)\) to obtain some results on singular \(M\)-type product s of distributions. He defines first the associated distribution \(u\) to an \(f\in G\) \((f\approx u)\), if it exists. The \(M\)-type product of two distributions \(u\) and \(v(u\cdot v)\) existe if the product of their embeddings into \(G\), \(\widetilde u\cdot\widetilde v\), admits the associated distribution \(w\), \(\widetilde u\cdot\widetilde v\approx w\). Then by definition \(u\cdot v=w\). The author of the paper proceeds to evaluate different singular \(M\)-type products of distributions: \(x^a_+\), \(x^a_-\) and \(\delta^{(p)}\), where \(a=(a_1, \dots,a_m)\), \(a_i\neq -1\), \(-2,\dots, i=1,\dots,m\) and \(p\in N^m_0\).