×

zbMATH — the first resource for mathematics

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.)
PDF BibTeX XML Cite
Full Text: DOI arXiv EuDML
References:
[1] V. Chistyakov: The Colombeau generalized nonlinear analysis and the Schwartz linear distribution theory. J. Math. Sci. 93 (1999), 42-133. · Zbl 0951.46018
[2] J.-F. Colombeau: New generalized functions. Multiplication of distributions. Physical applications. Contribution of J. Sebasti?o e Silva. Portugal Math. 41 (1982), 57-69. · Zbl 0599.46056
[3] J.F. Colombeau: New Generalized Functions and Multiplication of Distributions. North Holland Math. Studies 84, Amsterdam, 1984.
[4] B. Damyanov: Results on Colombeau product of distributions. Comment. Math. Univ. Carolinae 38 (1997), 627-634. · Zbl 0937.46030
[5] B. Damyanov: Mikusinski type products of distributions in Colombeau algebra. Indian J. Pure Appl. Math. 32 (2001), 361-375. · 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. · Zbl 0115.33101
[8] L. H?rmander: Analysis of LPD Operators I. Distribution Theory and Fourier Analysis. Springer-Verlag, Berlin, 1983.
[9] J. Jel?nek: Characterization of the Colombeau product of distributions. Comment. Math. Univ. Carolinae 27 (1986), 377-394. · Zbl 0612.46034
[10] J. Mikusinski: On the square of the Dirac delta-distribution. Bull. Acad. Pol. Ser. Sci. Math. Astron. Phys. 43 (1966), 511-513. · Zbl 0163.36404
[11] M. Oberguggenberger: Multiplication of Distributions and Applications to PDEs. Longman, Essex, 1992. · Zbl 0818.46036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.