# Article

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Keywords:
Schwartz distributions; multiplication; Colombeau generalized functions
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 Jan Mikusiński in 1966. 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.
References:
[1] V.  Chistyakov: The Colombeau generalized nonlinear analysis and the Schwartz linear distribution theory. J.  Math. Sci. 93 (1999), 42–133. DOI 10.1007/BF02365214 | MR 1683831 | Zbl 0951.46018
[2] J.-F.  Colombeau: New generalized functions. Multiplication of distributions. Physical applications. Contribution of J. Sebastião e Silva. Portugal Math. 41 (1982), 57–69. MR 0766839 | Zbl 0599.46056
[3] J. F.  Colombeau: New Generalized Functions and Multiplication of Distributions. North Holland Math. Studies 84, Amsterdam, 1984. MR 0738781 | Zbl 0532.46019
[4] B. Damyanov: Results on Colombeau product of distributions. Comment. Math. Univ. Carolinae 38 (1997), 627–634. MR 1601668 | Zbl 0937.46030
[5] B. Damyanov: Mikusiński type products of distributions in Colombeau algebra. Indian J.  Pure Appl. Math. 32 (2001), 361–375. MR 1826763 | Zbl 1021.46032
[6] I.  Gradstein and I.  Ryzhik: Tables of Integrals, Sums, Series, and Products. Fizmatgiz Publishing, Moscow, 1963.
[7] I.  Gel’fand and G.  Shilov: Generalized Functions, Vol. 1. Academic Press, New York and London, 1964. MR 0166596
[8] L.  Hörmander: Analysis of LPD  Operators  I. Distribution Theory and Fourier Analysis. Springer-Verlag, Berlin, 1983. MR 0717035
[9] J. Jelínek: Characterization of the Colombeau product of distributions. Comment. Math. Univ. Carolinae 27 (1986), 377–394. MR 0857556
[10] J.  Mikusiński: On the square of the Dirac delta-distribution. Bull. Acad. Pol. Ser. Sci. Math. Astron. Phys. 43 (1966), 511–513. MR 0203392
[11] M.  Oberguggenberger: Multiplication of Distributions and Applications to PDEs. Longman, Essex, 1992.

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