АПРОКСИМАТИВНІ ВЛАСТИВОСТІ УЗАГАЛЬНЕНИХ ІНТЕГРАЛІВ ПУАССОНА НА КЛАСАХ ФУНКЦІЇ, ЯКІ ВИЗНАЧАЮТЬСЯ ЗА ДОПОМОГОЮ МОДУЛЯ НЕПЕРЕРВНОСТІ

• Date: 2025
• Main Author: Kharkevych, Yu.I.
• Format: Article
• Language: English
• Published: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2025
• Subjects: модуль неперервності; асимптотична рівність; клас Гельдера; задача Колмогорова–Нікольського
• Online Access: https://jais.net.ua/index.php/files/article/view/637
• Journal Title: Problems of Control and Informatics

One of the most important problems of applied mathematics is the study of various problems of natural science, which ultimately leads to the compilation of mathematical models of the phenomena under study. Moreover, these mathematical models willbe of practical interest if and only if these models adequately reflect real situations. Often the objects studied are extremely complex. In such cases, some other method of obtaining additional information about this value may be a real find, allowing it to be found at least approximately. In this position, it is advisable to use the methods and approaches of the theory of approximation of functions, namely, the asymptotic estimates. The theory of approximation of functions is important because it provides general grounds for the practical calculation of functions, for the approximate replacement of complex functions by simpler ones. In this case, an important role is played by the modulus of continuity, which characterizes the maximum absolute increment of the function under study between the points of the domain of definition. Also important are the classes of functions that are defined by the modulus of continuity, in particular, the Holder classes. In this paper, we study the problem of finding the exact upper border of deviation of functions classes that are determined by a first order modulus of continuity, from their generalized Poisson integrals. In a partialcase, the asymptotic equalities were obtained for an approximation of functions from the Hölder classes by their generalized Poisson integrals. Thereby it is shown, that a transition from classes H ω to the more susceptible Hölder classes H 1 provides more qualitative solution of the Kolmogorov–Nikol’skii problem for generalized Poisson integrals in the uniform metric, that has a direct applicationin mathematical modeling and in mathematical formalizations in certain types of problems in game theory.