НАБЛИЖЕННЯ СПРЯЖЕНИХ ПЕРІОДИЧНИХ ФУНКЦІЙ ЇХ ТРИГАРМОНІЙНИМИ ІНТЕГРАЛАМИ ПУАССОНА

• Datum: 2020
• 1. Verfasser: Hrabova , U.Z.
• Format: Artikel
• Sprache: English
• Veröffentlicht: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2020
• Schlagworte: полігармонічні рівняння; механіка суцільних середовищ; асимптотичні рівності; спряжені функції
• Online Zugang: https://jais.net.ua/index.php/files/article/view/495
• Назва журналу: Problems of Control and Informatics

Asymptotic equalities are obtained for the upper bounds of deviations of the three-harmonic Poisson integrals from functions  conjugated to functions from the Hölder classes   in the uniform metric. Thus, one of the most important problems is solved in the theory of approximation of functions — the Kolmogorov–Nikol’skii problem for the class   and three-harmonic Poisson integral in the uniform metric, that eliminates the gap in the solution of this problem for classes of conjugated periodic functions. In the paper the methods of investigation are used for the integral representations of deviations of operators defined by the sequence of functions  that depend on a certain continuous parameter  on the classes of periodic functions. They arose and developed due to the papers by L.I. Bausov. The classical theory of boundary problems for polyharmonic functions became a well-organized chapter of the mathematical modeling. Modeling of phenomena studied in mechanics of continuous media (a plane problem of elasticity theory, the elasticity problem of a thin plate with rigid edges etc.) leads to boundary value problems for polyharmonic equation in the specific area. Since the development of high-precision production leads to the need for development and implementation of asymptotic methods of approximation theory, then the results of this work can be considered as possible applications. This is due to the fact that asymptotic methods are much more constructive and simpler in computational implementation than the exact solution methods (in case there are any). In the real conditions (especially when developing precision engineering software) asymptotic methods lead to almost the same results as optimal. As a result of this study, we can conclude that with further improvement of technology towards high-precision production, namely, asymptotic methods will be given a priority. This is facilitated by the development of science-intensive, high-precision technologies and the recent success in the development of extreme problems of the approximation theory. In addition, a lot of technical problems of modern engineering generate new formulations of problems in the theory of approximation.