Uniform approximation of Poisson integrals of functions from the class H_omega by de la Vallee Poussin sums

TL;DR

This paper derives asymptotic bounds for the uniform approximation errors of Poisson integrals of functions with specific smoothness properties using de la Vallée Poussin sums, addressing a classical approximation problem.

Contribution

It provides new asymptotic equalities for approximation errors of Poisson integrals by de la Vallée Poussin sums, extending results to functions with convex moduli of continuity.

Findings

Asymptotic equalities for deviation bounds are established.

Results solve the Kolmogorov-Nikol'skii problem for certain function classes.

Approximation errors are characterized for functions with specific smoothness.

Abstract

We obtain asymptotic equalities for least upper bounds of deviations in the uniform metric of de la Vall\'{e}e Poussin sums on the sets C^{q}_{\beta}H_\omega of Poisson integrals of functions from the class H_\omega generated by convex upwards moduli of continuity \omega(t) which satisfy the condition \omega(t)/t\to\infty as t\to 0. As an implication, a solution of the Kolmogorov-Nikol'skii problem for de la Vall\'{e}e Poussin sums on the sets of Poisson integrals of functions belonging to Lipschitz classes H^\alpha, 0<\alpha <1, is obtained