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multiplication of Schwartz distributions; Colombeau generalized functions
The differential $\Bbb C$-algebra $\Cal G(\Bbb R^m)$ of generalized functions of J.-F. Colombeau contains the space $\Cal D'(\Bbb R^m)$ of Schwartz distributions as a $\Bbb C$-vector subspace and has a notion of `association' that is a faithful generalization of the weak equality in $\Cal D'(\Bbb R^m)$. This is particularly useful for evaluation of certain products of distributions, as they are embedded in $\Cal G(\Bbb R^m)$, in terms of distributions again. In this paper we propose some results of that kind for the products of the widely used distributions $x_{\pm}^a$ and $\delta ^{(p)}(x)$, with $x$ in $\Bbb R^m$, that have coinciding singular supports. These results, when restricted to dimension one, are also easily transformed into the setting of regularized model products in the classical distribution theory.
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