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 $C_\beta^\psi L_s, \ 1\leq s\leq\infty$, generated by sequences $\psi(k)>0$ satisfying the condition d'Alembert $\mathop {\rm \lim}\limits_{k\rightarrow\infty}\frac{\psi(k+1)}{\psi(k)}=q, \ q\in[0,1)$, asymptotically unimprovable estimates for deviations of de la Vall\'{e}e 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.