АЛГОРИТМІЗАЦІЯ ОБЧИСЛЕНЬ КОНСТАНТ КОЛМОГОРОВА–НІКОЛЬСЬКОГО ВЕЛИЧИН НАБЛИЖЕННЯ СПРЯЖЕНИХ ДИФЕРЕНЦІЙОВНИХ ФУНКЦІЙ УЗАГАЛЬНЕНИМИ ІНТЕГРАЛАМИ ПУАССОНА

• Дата: 2025
• Автор: Zhyhallo, K.N.
• Формат: Стаття
• Мова: English
• Опубліковано: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2025
• Теми: оцінка похибки; класи спряжених функцій; чисельні алгоритми; математичне моделювання; оптимальне управління; ігрові задачі динаміки; оператори; необхідна точність наближення
• Онлайн доступ: https://jais.net.ua/index.php/files/article/view/678
• Назва журналу: Problems of Control and Informatics

In applied mathematics in solving a number of problems, it is advisable to use the methods and approaches of approximation theory. One of the most important types of problems, of both the theory of approximation of functions and applied mathematics,is the so-called extremal problems of Kolmogorov–Nikolʼskii. The essence of the Kolmogorov–Nikolʼskii problem in applied mathematics is the approximation of some mathematical objects by others, usually of a simpler nature, whose propertiesare already known, and the necessary characteristics are calculated in one way or another. In this case, an important role is played by the error estimate of the obtained approximation, which will directly depend on the accuracy of solving the Kolmogorov–Nikolʼskii problem. And this accuracy will directly depend on the number of terms in complete asymptotic expansions (by powers (1−p), p →1−0, in this article). The constants that face the corresponding degrees (1−p), p →1−0, in complete asymptotic expansions in applied mathematics are called the Kolmogorov–Nikolʼskii constants. Obviously, the more we know these Kolmogorov–Nikolʼskii constants, the more accurately we can get the degree of error when some mathematical objects are approximated by others. An algorithm has been developed for computing the Kolmogorov–Nikolʼskii constants of arbitrarily high order of smallness when approximating conjugate differentiable functions by their generalized Poisson integrals. The result obtained in this paper will allow us to expand significantly the boundaries of the application of problems of the theory of approximation in applied mathematics,namely, when constructing numerical algorithms, when considering optimal control problems, in mathematical modeling of complex technical and ecological systems, etc.