Lebesgue-type inequalities for de la Vallee Poussin sums on the sets of analytic and entire functions

Abstract

For the functions from sets $C_{\beta }^{\psi }C$ and $C_{\beta }^{\psi }{L}_s,1\le s\le \infty$, generated by sequences $\psi (k)>0$ satisfying the condition d'Alembert $\underset{k\to \infty }{\lim }\frac{\psi (k+1)}{\psi (k)}=q,q\in [0,1)$, asymptotically unimprovable estimates for deviations of de la Vallée Poussin sums in the uniform metric, which are represented in terms of values of the best approximations of $(\psi ,\beta )$-differentiable functions of this sort by trigonometric polynomials in the metrics $L_s$ are obtained. Proved that received estimates are unimprovable on some important functional subsets.