Open AccessMultiplication of Distributions and Algebras of Mnemofunctions
Author(s) -
А. Б. Антоневич,
А. Б. Антоневич,
Т. Г. Шагова,
Т. Г. Шагова
Publication year - 2019
Publication title -
sovremennaâ matematika. fundamentalʹnye napravleniâ
Language(s) - English
Resource type - Journals
eISSN - 2949-0618
pISSN - 2413-3639
DOI - 10.22363/2413-3639-2019-65-3-339-389
Subject(s) - multiplication (music) , mathematics , embedding , distribution (mathematics) , operator (biology) , space (punctuation) , algebra over a field , pure mathematics , order (exchange) , function (biology) , mathematical analysis , computer science , combinatorics , biochemistry , chemistry , finance , repressor , artificial intelligence , evolutionary biology , biology , transcription factor , economics , gene , operating system
In this paper, we discuss methods and approaches for denition of multiplication of distributions, which is not dened 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 eects arising in study of equations with distributions as coecients are demonstrated by example of a linear rst-order dierential equation.