Multiplication of Distributions and Algebras of Mnemofunctions

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.

Extensions of nonclosable operators and multiplication of distributions

In the paper some new constructions of extensions of nonclosable operators is proposed and several examples of applications are given. One of particular cases of the problem under consideration is the question on multiplication of distributions, a solution to which can be given by introduction of the so-called new generalized functions. It was demonstrated that the main obstacle for multiplication of distributions is nonclosablility of classical multiplication and the construction of new generalized functions is based on the ideas similar to that used under construction of the extension of nonclosable operators.