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

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.