Approximation properties of Abel-Poisson integrals on the classes of differentiable functions, defined by means of modulus of continuity

Abstract

Being the natural apparatus of the periodic functions approximation, the partial Fourier sums are not uniformly convergent over the entire space of the continuous functions. This fact stimulated the search for ways to construct sequences of polynomials that would converge uniformly on the entire space. The matrix method of Fourier series summation is one of the most common methods. Many results on the approximation of the classes of differentiated functions have been obtained for methods generated by triangular infinite matrices. The set of approximating linear methods can be extended by the process of summation of Fourier series, when instead of an infinite triangular matrix one considers the set $\Lambda=\{\lambda_{\delta}(k)\}$ of functions of the natural argument depending on the real parameter $\delta$. The paper deals with the problem of approximation in the uniform metric of $W^{1}H_{\omega}$ classes using one of the classical linear summation methods for Fourier series given by a set of functions of a natural argument, namely, using the Abel-Poisson integral. At the same time, emphasis is placed on the study of the asymptotic behavior of the exact upper limits of the deviations of the Abel-Poisson integrals from the functions of the mentioned class.