The New Approximation Effects of Weyl-Nagy Kernels

• Дата: 2021
• Автори: Сорич, Віктор, Сорич, Ніна
• Формат: Стаття
• Мова: Українська
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2021
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/251178
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

In a uniform metric, the problem of obtaining the exact values of the best approximations on classes 2π-periodic functions, r-th (r > 0) derivatives of which are in a single sphere of space of significantly limited functions, there was decided in 1936 by J. Favar [1]. Such classes can also be considered as classes of convolutions with known in the scientific literature on the theory of approximation Bernoulli kernels. In solving the problem J. Favar put forward the hypothesis that the similar problem with fractional values of the parameter r can also be solved according to the proposed scheme. The idea of solving the problem is based on Rolle's theorem on the relationship between the number of zeros of a function and the number of zeros of its derivative. Lately we have been seeing the increased attention of mathematics to the problems, for which Rolle’s theorem is true, and as result of which it becomes possible to find solutions to many problems of the theory of approach. Many outstanding mathematicians worked on J. Favar's hypothesis: N. Akhiezer, M. Krein, S. Nikolsky, S. Stechkin, Sun Yon-shen and others. The final results for solving the problem of finding the exact values of the best approximations in the classes, generated by the Weyl-Nagy kernels and which generalize Bernoulli kernels, in the metrics of spaces of continuous and according of summary functions, belong to V. Dzyadyk [2].  The problem of joint approximation of periodic functions and their derivatives in the formulation, which is considered in this paper, was initiated by O. Stepanets. The finding the exact value of the sizes of the best approximations of the individual, and the most important (according to the successful proposal of O. Stepanets [3]) linear combinations functions from Weyl-Nagy classes in uniform and integral metrics was considered in the works of the authors in detail (see, in particular, [4, 5]). This works concern the best joint approximation of functions from classes of convolutions with fixed generating kernels. If the number of terms m in a linear combination is one, then the best joint approximation and value of the best approximations coincide. This article is a logical continuation of a problem of finding the values of the best joint approximation of linear combinations of functions from Weyl-Nagy classes in metrics of spaces of continuous and, accordingly, summary functions; the values of the parameters that complements the before considered cases is investigated in it. It found the conditions for parameters of the problem of the best joint approximation, in which the kernels of convolution satisfy Nagy sufficient conditions of best approximation in the integral metrics.