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 $\phi$ from $L_p$, $1\le p<\infty$. In these inequalities uniform norms of deviations of Fourier sums $\Vert f-S_{n-1}{\Vert }_C$ are expressed via best approximations $E_n(\phi {)}_{L_p}$ of functions $\phi$ by trigonometric polynomials in the metric of space $L_p$. We show that obtained estimates are asymptotically best possible.

15 pages