Best Approximation in the Integral Metric of a Linear Combination of Periodic Functions of Different Parity
Authors:   Kamianets-Podіlskyi Ivan Ohiienko National University, Sorych Viktor , Sorych Nina ,   Kamianets-Podіlskyi Ivan Ohiienko National University
Published: October 2024

In many areas of mathematics, extremal problems often arise related to approximation characteristics of both the approximating functions and the properties of the elements being approximated. For example, in the case of polynomials, the problem can involve the number of points where the values of the function coincide with the values of the polynomial used to replace the function on the studied interval. In practice, the problem of approximating a function from a given set R is reduced to replacing it, according to a defined algorithm, with a function (polynomial) of a fixed degree that is close to it in a certain sense. In uniform and integral metrics, the problem of finding the exact values of the best approximations for classes of r-times differentiable functions, where r is a natural number, has been explored in the works of J. Favard [1], N. I. Akhiezer, M. G. Krein [2], B. Nadi, S. M. Nikolsky [3], V. K. Dzyadyk [4], S. B. Stechkin, Sun Yun-shen, and others. The final results concerning the solution of the best approximation problem on Weyl-Nagy classes for arbitrary values of the parameters defining these classes belong to the Ukrainian mathematician V. K. Dzyadyk [4]. In his works on function classes generated by the well-known Bernoulli kernels, it was established that the number of coincidence points between the kernel and the approximating polynomial of degree n − 1, including their multiplicities, does not exceed 2n, which allowed for obtaining final results. The work [5] presents cases of such linear combinations of even or odd kernels for which the number of uniformly distributed interpolation points equals 2n + 2 for a polynomial of degree n − 1 that deviates the least in the metric of the L – space from the studied linear combination. The idea of studying composite kernels expressed as a linear combination of component terms belongs to O. I. Stepanyets [6], and it was implemented in problems of joint approximation of functions and their derivatives. In the 1980s and 1990s, O. I. Stepanets developed a new approach to classifying periodic functions, which allowed for a fine classification of extremely broad sets of periodic functions. The results obtained for these classes are, on one hand, of a general nature, and on the other, they provide a whole series of new, previously unknown results that could not be achieved with previously known classes. Following the approaches to function classification, we can consider a linear combination of function classes as a certain single class of a more complex nature. Then, the problem of finding the exact values of the upper bounds of the best approximations reduces to the problem of the best approximation for this composite class, which corresponds to convolutions with the generating composite kernel. In this work, we investigate linear combinations of three continuous 2π-periodic kernels of different parity, and it is established that there exist such composite kernels that their trigonometric polynomial of the best approximation of order n − 1 in the integral metric interpolates the kernel at 2n + 2 uniformly distributed points