The Inverse Theorem for the Generalized Derivative in Banach Spaces

• Main Authors: Радзієвська, Олена, Ковальська, Ірина
• Format: Article
• Language: Ukrainian
• Published: Кам'янець-Подільський національний університет імені Івана Огієнка 2023
• Online Access: http://mcm-math.kpnu.edu.ua/article/view/296478

Establishing the properties of the approximation characteristics of the studied functions is one of the main tasks of the theory of approximations. If, based on information about the behavior of the generalized derivative of some function f, it is possible to predict the behavior of the sequence of the best approximations of this function by polynomials, then we are talking about stating and proving direct theorems of the theory of approximations. If the properties of the function f Î X itself and its generalized derivatives are studied, relying on the behavior of the sequence best approximations, i.e., the differential-difference characteristics of the function f are established based on the study of the behavior of the sequence of its best approximations, then we speak of the proof of inverse theorems of approximation theory. The study of direct and inverse theorems begins with the works of Bernstein, Valle Poussin, Jackson and others in 1910-1912. They were continued by many scientists (N. I. Ahiezer, M. G. Crane, J. Favar, B. V. Stechkin, S. M. Nikolskyi, A. F. Timan, A. Zygmund, V. K. Dzya­dyk, O. I. Stepanets). There are still many important and unsolved problems in the theory of approximations, in particular, such as extending direct and inverse theorems to new classes of functions and establishing the best values of constants in the corresponding inequalities. At the same time, it becomes possible to formulate new problems, in particular, problems of mathematical modeling already for whole classes of functions, that describe the studied processes. This article considers the inverse theorem – based on the properties of the sequence of best approximations, a conclusion is made about the properties of the element f of some Banach space X and its generalized derivatives. As well as the relations between Szego constants for different equivalent systems of elements of the Banach space are established