Approximation of analytic periodic functions by linear means of Fourier series

The work concerns the questions of approximation of periodic differentiable functions of high smoothness by repeated arithmetic means of Fourier sums. One of the classifications of periodic functions nowadays is the classification suggested by A. Stepanets which is based on the concept of $(\psi,\beta)$-differentiation. The given classification allows to distinguish all classes of summable periodic functions from the functions where the Fourier series can deviate to infinitely differentiable functions including analytical and entire ones. When choosing the parameters properly, classes of $(\psi,\beta)$-differentiable functions exactly coincide with the well-known classes of Vail differentiable functions, Sobolev classes $W^l_p$ and classes of convolutions with integral kernels.
De la Vallee Poussin sums and their special cases (Fourier sums and Fejer sums) were extensively studied for many decades by many prominent experts in the theory of functions. At present, the large amount of factual material is accumulated in numerous publications. One of the most important directions in this field is the investigation of approximation properties of these sums for various classes of functions. The aim of the present paper is to systematize the known results related to the approximation properties of de la Vallee Poussin sums on classes of Poisson integrals and to present new facts obtained for their generalizations.
In certain cases asymptotic equalities are found for upper bounds of deviations in the uniform metric of the repeated de la Vallee Poussin sums on the classes $C^\psi_{\beta,\infty}$ and $C^\psi_\beta H_\omega$ which are generated by multiplicators $\psi(k)$ and shifts on argument $\beta$ provided that sequences $\psi(k)$ which define the specified classes tend to zero with the rate of geometrical progression. In doing so classes $C^\psi_{\beta,\infty}$ and $C^\psi_\beta H_\omega$ consist of analytic functions which can be regularly extended in the corresponding strip.
We introduce generalized de la Vallee Poussin sums and study their approximation properties for the classes of analytic periodic functions. We obtain asymptotic equalities for upper bounds of deviations of the repeated de la Vallee Poussin sums on classes of Poisson integrals. Under certain conditions, these equalities guarantee the solvability of the Kolmogorov–Nikol’skiy problem for the repeated de la Vallee Poussin sums and classes of Poisson integrals. We indicate conditions under which the repeated sums guarantee a better order of approximation than ordinary de la Vallee Poussin sums.
Bibliography: 16 titles.