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 \(Z^{s}_{n}\) of classes \(C^{\psi}_{\beta,p}\) 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 \(\Psi_{\beta}\in L_{p'}\), \(\frac{1}{p}+\frac{1}{p'}=1\). In addition, we find a set of allowed values of parameters (that define the class \(C^{\psi}_{\beta,p}\) and the linear method \(Z^{s}_{n}\)) for which Zygmund sums and Fejer sums realize the order of the best uniform approximations by trigonometric polynomials of those classes.