Estimation of the Best Approximations for the Generalized Derivative in Banach Spaces

• Дата: 2021
• Автори: Радзієвська, Олена, Ковальська, Ірина
• Формат: Стаття
• Мова: English
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2021
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/251176
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

The main task of the theory of approximation is to establish the properties of the approximation characteristics of this function on the basis of its investigated properties. Functions with the same properties are grouped into classes, and then the facts established for a particular class apply to each of its representatives. This makes it possible to formulate new problems, in particular mathematical modeling problems for whole classes of functions that describe the studied processes. If the statements allow on the basis of information about the generalized derivative of the element f to draw a conclusion about the rate of approach to zero of the sequence of the best approximations of this element by polynomials of degree n, then in the theory of approximations they are called direct theorems. In the given article the inverse theorem is considered — per properties of sequence of the best approximations we draw a conclusion about properties of an element f of some Banach space B and its generalized derivatives. That is, according to a given sequence of the best approximations of the vector f by polynomials of degree n establish its differential characteristics. The first inverse theorems were considered at the beginning of the last century by S. N. Bernstein. The main point of their proof is the inequalities between the norms of polynomials and their derivatives. Such inequalities are called Bernstein inequalities. As a partial case, they can be obtained from the theorem considered in the article.