ПОВНІ АСИМПТОТИКИ НАБЛИЖЕНЬ ДЕЯКИМИ СИНГУЛЯРНИМИ ІНТЕГРАЛАМИ В МАТЕМАТИЧНОМУ МОДЕЛЮВАННІ

• Дата: 2020
• Автор: Zhyhallo, K.N.
• Формат: Стаття
• Мова: English
• Опубліковано: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2020
• Теми: математичне моделювання; ігрові задачі динаміки; класи Ліпшиця; сингулярні інтеграли
• Онлайн доступ: https://jais.net.ua/index.php/files/article/view/508
• Назва журналу: Problems of Control and Informatics

In solving some types of applied problems, the most effective nowadays are methods of the theory of approximation of functions. In a modern stage of development of the theory of approximation of functions, one merely deals with either an approximation of individual functions or whole function classes by a preset subsets of functions that appear in a certain sense more convenient to deal with in calculations in comparison to the functions that should be approximated. In practice one often chooses a set of algebraic polynomials of a defined order as such subspace. As a result, a new type of problems appeared, that further was called the extremal problems of the theory of approximation. In turn, among all of the extremal problems of the theory of approximation the most interesting from the mathematical modelling point of view are the socalled Kolmogorov-Nikol’skii problems. The main goal of the KolmogorovNikol’skii problem is to find the asymptotic equalities for the values of the approximation of functions of certain classes of specific methods of summation of Fourier series. In the paper a problem is considered of an approximation of 2ï -periodic functions from the Lipshitz class by certain singular integrals. The most prominent examples of such integrals are the so-called generalized Poisson integrals. As a result, we wrote down the complete asymptotic expansions in terms of 1 , d ® 00 , of the least d upper borders of deviations of functions from the Lipshitz class from their generalized Poisson integrals. The obtained result allows us to write down not only the main term of the asymptotic expansion, but also using the Riemann z -function write down its second, third terms, etc., that, respectively, much simplifies the problem of algorithmization in solving of the stated applied problem. Moreover, the generalized Poisson integrals are the solutions of partial differential equations they are directly connected to the methods of solving of integral, difference-differential and integrodifferential games, that are related to the game problems of dynamics.