ON LINEAR SUBSTITUTION OF VARIABLES AND ON SIMPLIFICATION OF EXPRESSIONS IN THE SPACE OF GENERALIZED FUNCTIONS

Abstract

The rigorous mathematical theory of generalized functions contains the definition of a set of basic functions, the definition of continuous functional, rules for performing limit transitions, delta-like sequences, etc. Special literature is devoted to these questions, in which one can find formulations and proofs of the corresponding theorems (see, for example, the book by V.S.Vladimirov “Generalized functions in mathematical physics”, I.M.Gelfand and G.E.Shilov “Generalized functions and actions on them”, M.S.Agranovich “Generalized functions and Sobolev spaces”, M.A.Shubin “Lectures on the equations of mathematical physics” and many others). In this article, we will look at techniques for substitution a variable in an argument of generalized function, multiplying by a local-integrable function, and simplifying expressions. The theory of generalized functions has been chosen as a mathematical apparatus. The technique of generalized functions provides a convenient apparatus for solving a number of classical problems of an applied nature. The proposed article will be useful for students of physics and mathematics, engineering and physics specialties of higher educational institutions, as well as master students, teachers of relevant universities, studying and interested in the practical application of the theory of the generalized.