Estimates of the uniform approximations by Zygmund sums on the classes of convolutions of periodic functions

Abstract

We obtain order-exact estimates for uniform approximations by using Zygmund sums $Z_n^s$ of classes $C_{\beta ,p}^{\psi }$ of $2\pi$-periodic continuous functions $f$ representable by convolutions of functions from unit balls of the space $L_p$, $1<p<\infty$, with a fixed kernels ${\Psi }_{\beta }\in L_{p^{\prime }}$, $\frac{1}{p}+\frac{1}{p^{\prime }}=1$. In addition, we find a set of allowed values of parameters (that define the class $C_{\beta ,p}^{\psi }$ and the linear method $Z_n^s$) for which Zygmund sums and Fejer sums realize the order of the best uniform approximations by trigonometric polynomials of those classes.

arXiv comment17 pages, in Ukrainian