Asymptotically best possible Lebesque-type inequalities for the Fourier sums on sets of generalized Poisson integrals

TL;DR

This paper derives optimal Lebesgue-type inequalities for Fourier sums of functions defined by generalized Poisson integrals, relating deviations to best approximations and proving their asymptotic optimality.

Contribution

It establishes the asymptotically best possible Lebesgue-type inequalities for Fourier sums of generalized Poisson integral functions, linking deviations to best approximations in Lp spaces.

Findings

Derived Lebesgue-type inequalities for Fourier sums.

Proved the asymptotic optimality of these inequalities.

Connected deviations of Fourier sums to best approximation measures.

Abstract

In this paper we establish Lebesgue-type inequalities for $2\pi$-periodic functions $f$, which are defined by generalized Poisson integrals of the functions $\varphi$ from $L_{p}$, $1\leq p< \infty$. In these inequalities uniform norms of deviations of Fourier sums $\| f-S_{n-1} \|_{C}$ are expressed via best approximations $E_{n}(\varphi)_{L_{p}}$ of functions $\varphi$ by trigonometric polynomials in the metric of space $L_{p}$. We show that obtained estimates are asymptotically best possible.