Approximation of classes of Poisson integrals by Fejer means

Abstract

The paper is devoted to the investigation of problem of approximation of continuous periodic functions by trigonometric polynomials, which are generated by linear methods of summation of Fourier series.

The simplest example of a linear approximation of periodic functions is the approximation of functions by partial sums of their Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. Therefore, many studies devoted to the research of the approximative properties of approximation methods, which are generated by transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for the whole class of continuous functions. Particularly, Fejer sums have been widely studied recently. One of the important problems in this area is the study of
asymptotic behavior of the sharp upper bounds over a given class of functions of deviations of the trigonometric polynomials.

In the paper, we study upper asymptotic estimates for deviations between a function and the Fejer means for the Fourier series of the function. The asymptotic behavior is considered for the functions represented by the Poisson integrals of periodic functions of a real variable. The mentioned classes consist of analytic functions of a real variable. These functions can be regularly extended into the corresponding strip of the complex plane.
An asymptotic equality for the upper bounds of Fejer means deviations on classes of Poisson integrals was obtained.