Exact Constants of the Best One Sided Approximations of the Sum Analytic Functions from Different Classes

• Date: 2022
• Main Authors: Сорич, Віктор, Сорич, Ніна
• Format: Article
• Language: Ukrainian
• Published: Кам'янець-Подільський національний університет імені Івана Огієнка 2022
• Online Access: http://mcm-math.kpnu.edu.ua/article/view/274097
• Journal Title: Mathematical and computer modelling. Series: Physical and mathematical sciences

The task of obtaining the exact values of the best approximations by trigonometric polynomials of continuous or summable functions originates from the works of P. L. Chebyshev, who, back in the 1950s, posed the problem of finding the polynomial, that deviates the least from a given continuous function. Subsequently this direction in the theory of approximation got further development thanks to the works of K. Weierstrass, D. Jackson, S. N. Bernstein, Valle-Poussin and others. On this time there is an increase attention to problems of one-sided approximation of individual functions and their classes in the metric space L. Problems of this content arise up in number theory, coding theory, and other areas of mathematics. The first results of this direction were obtained in the 1880th by A. A. Markov and T. Y. Stieltijes. In the future, these studies were continued in the works of J. Karamata (1930), G. Freud and T. Hanelius (mid-20th century). General issues related to the problem of the best approximation of classes of functions by trigonometric polynomials: the existence of the best approximation polynomial, its characteristic properties, one sided approximations are detailed in many works, in particular, for example, in the book of M. P. Korneychuk [1], the works of T. Ganelius [4] , V. G. Doronin, A. A. Ligun [5]. The exact constants of the best one sided approximation of the sum of the majorant functions of the classes that allow analytical extension into a strip of fixed width and of functions harmonic in a circle of radius 1 have been found in this work.