On approximation of classes of analytic periodic functions by Fejer means

The paper is devoted to the approximation of periodic functions of high smoothness by arithmetic means of Fourier sums. The simplest and natural example of a linear process of approximation of continuous periodic functions of a real variable is the approximation of these functions by partial sums of the Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the entire class of continuous $2\pi$-periodic functions. In connection with this, a significant number of papers is devoted to the study of the approximative properties of other approximation methods, which are generated by certain transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for each function $f \in C$. In particular, over the past decades, de la Vallee Poussin sums and Fejer sums have been widely studied. Today, publications have accumulated a large amount of factual material. One of the most important directions in this field is the study of the asymptotic behavior of upper bounds of deviations of arithmetic means of Fourier sums on different classes of periodic functions. Methods of investigation of integral representations of deviations of trigonometric polynomials generated by linear methods of summation of Fourier series, were originated and developed in the works of S.M. Nikolsky, S.B. Stechkin, N.P. Korneichuk, V.K. Dzadyk and others.
The aim of the work is to systematize known results related to the approximation of classes of periodic functions of high smoothness by arithmetic means of Fourier sums and to present new facts obtained for particular cases and to present new approximative properties of Fejer sums on the classes of periodic functions that can be regularly extended into the corresponding strip of the complex plane. Under certain conditions, we obtained asymptotic formulas for upper bounds of deviations in the uniform metric of Fejer sums on Poisson integrals classes. The deduced formula provides a solution of the corresponding Kolmogorov-Nikolsky problem without any additional conditions.