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Balanced Colombeau products of the distributions $$x_{\pm }^{-p}$$ and $$x^{-p}$$. (English) Zbl 1081.46027
Summary: Results on singular products of the distributions $$x_{\pm }^{-p}$$ and $$x^{-p}$$ for natural $$p$$ are derived when the products are balanced, so that their sum exists in the distribution space. These results follow the pattern of a known distributional product published by J. Mikusiński in [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 14, 511–513 (1966; Zbl 0163.36404)]. The results are obtained in the Colombeau algebra of generalized functions, which is the most relevant algebraic construction for tackling nonlinear problems of Schwartz distributions.

##### MSC:
 46F10 Operations with distributions and generalized functions 46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
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##### References:
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