ПРО МАТРИЧНЕ ПІДСУМОВУВАННЯ РЯДІВ ФУР’Є

• Дата: 2020
• Автори: Voloshyna, T.V., Baysalov, Zh.U.
• Формат: Стаття
• Мова: English
• Опубліковано: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2020
• Теми: диференціальні рівняння в частинних похідних; ігрові задачі динаміки; ряди Фур’є; лінійні методи підсумовування
• Онлайн доступ: https://jais.net.ua/index.php/files/article/view/507
• Назва журналу: Problems of Control and Informatics

One of the most important problems of applied mathematics is the series summation, in particular, linear matrix summation of the Fourier series. This problem appears, e.g., in finding the best approximation as well as in solving game problems of dynamics, which even more highlights a necessity of the detailed investigation of linear matrix methods of summation of the Fourier series. Systematic research on the theory of summation of Fourier series using triangular matrices began under the influence of the works of B. Nagy, S.M. Nikol’skii, S.A. Telyakovskii and other mathematicians in the middle of the 20th century and are actively continuing in our time. If we talk about studies of rectangular linear methods of summation of the Fourier series, then many questions remain open. Therefore, in this work we consider the basic theoretical concepts in the field of linear rectangular matrix methods of summation of the Fourier series, such that are little discussed in scientific, including foreign, literature. The investigations gave us possibility to formulate advantages of linear rectangular matrix methods of summation of the Fourier series over linear triangular matrix methods of summation in the sense that first are applicable to solving game problems of dynamics. The problem considered in this paper for an n-harmonic equation in the unit circle enables us to write down successively specific linear rectangular methods of summation of the Fourier series, namely, the Abel–Poisson method, the Weierstrass method, and the methods of biharmonic and three-harmonic Poisson integrals. Moreover, each of the mentioned above linear rectangular matrix methods of summation of the Fourier series is a solution of a corresponding partial differential equation (elliptic-type). The latest novation could significantly expand the class of game problems of dynamics, such that can be investigated taking into account the theoretical aspects of linear rectangular matrix methods of summation of the Fourier series that are described in this paper.