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 CMFD, 2019, Volume 65, Issue 3, Pages 339–389 (Mi cmfd383)

Multiplication of distributions and algebras of mnemofunctions

A. B. Antonevich, T. G. Shahava

Belarusian State University, Minsk, Belarus

Abstract: In this paper, we discuss methods and approaches for definition of multiplication of distributions, which is not defined in general in the classical theory. We show that this problem is related to the fact that the operator of multiplication by a smooth function is nonclosable in the space of distributions. We give the general method of construction of new objects called new distributions, or mnemofunctions, that preserve essential properties of usual distributions and produce algebras as well. We describe various methods of embedding of distribution spaces into algebras of mnemofunctions. All ideas and considerations are illustrated by the simplest example of the distribution space on a circle. Some effects arising in study of equations with distributions as coefficients are demonstrated by example of a linear first-order differential equation.

DOI: https://doi.org/10.22363/2413-3639-2019-65-3-339-389

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UDC: 517.9

Citation: A. B. Antonevich, T. G. Shahava, “Multiplication of distributions and algebras of mnemofunctions”, Proceedings of the Crimean autumn mathematical school-symposium, CMFD, 65, no. 3, PFUR, M., 2019, 339–389

Citation in format AMSBIB
\Bibitem{AntSha19} \by A.~B.~Antonevich, T.~G.~Shahava \paper Multiplication of distributions and algebras of mnemofunctions \inbook Proceedings of the Crimean autumn mathematical school-symposium \serial CMFD \yr 2019 \vol 65 \issue 3 \pages 339--389 \publ PFUR \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd383} \crossref{https://doi.org/10.22363/2413-3639-2019-65-3-339-389}