Lebesque-type inequalities for the Fourier sums on classes of generalized Poisson integrals

TL;DR

This paper derives optimal Lebesgue-type inequalities for Fourier sums of generalized Poisson integrals, providing precise bounds on deviations in uniform metric based on best approximations of derivatives.

Contribution

It introduces new asymptotically optimal inequalities for Fourier sums of generalized Poisson integrals in the uniform metric.

Findings

Derived upper bounds for deviations of Fourier sums

Established asymptotic optimality of the estimates

Connected deviations to best approximations of derivatives

Abstract

For functions from the set of generalized Poisson integrals $C^{\alpha,r}_{\beta}L_{p}$, $1\leq p <\infty$, we obtain upper estimates for the deviations of Fourier sums in the uniform metric in terms of the best approximations of the generalized derivatives $f^{\alpha,r}_{\beta}$ of functions of this kind by trigonometric polynomials in the metric of the spaces $L_{p}$. Obtained estimates are asymptotically best possible.