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

TL;DR

This paper derives near-optimal bounds for de la Vallée Poussin sums approximating certain analytic and entire functions, expressed via best approximation measures, and proves their optimality on key function classes.

Contribution

It provides asymptotically unimprovable Lebesgue-type inequalities for de la Vallée Poussin sums on specific function sets, linking approximation errors to best approximation metrics.

Findings

Derived asymptotically optimal deviation estimates in uniform metric.

Expressed deviations in terms of best approximations by trigonometric polynomials.

Proved the estimates are unimprovable on important functional subsets.

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.