Lebesgue-type inequalities for de la Vallee Poussin sums and their interpolation analogues on the sets $(\psi,\bar{\beta})$-differentiable functions

TL;DR

This paper establishes Lebesgue-type inequalities for de la Vallée Poussin sums and their interpolation analogues, providing estimates for their deviation rates on classes of functions characterized by $(ar{eta})$-differentiability and best approximation measures.

Contribution

It introduces new Lebesgue-type inequalities for these sums and their analogues on classes of $(ar{eta})$-differentiable functions, linking deviations to best approximation measures.

Findings

Derived explicit deviation estimates for de la Vallée Poussin sums.

Extended inequalities to interpolation analogues of these sums.

Connected deviations with best approximation measures in $L_s$ spaces.

Abstract

We obtain the estimates of steady rates of deviations of the de Vall\'{e}e Poussin sums and interpolation analogues of sums of Vall\'{e}e Poussin from the functions that belong to the space $C_{\bar{\beta}}^\psi L_s, \ 1\leq s\leq\infty$ and are represented through the best approximations of $(\psi,\bar{\beta})$-differentiable functions of this sort by trigonometric polynomials in the metric $L_s$