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 2π-periodic functions f , which are defined by generalized Poisson integrals of the functions ϕ from L p , 1 ≤ p < ∞. In these inequalities uniform norms of deviations of Fourier sums f − S n−1 C are expressed via best approximations E n (ϕ) Lp of functions ϕ by trigonometric polynomials in the metric of space L p . We show that obtained estimates are asymptotically best possible. K e y w o r d s Lebesgue-type inequalities, Fourier sums, generalized Poisson integrals, best approximations by trigonometric polynomials

About Lebesgue inequalities on the classes of generalized Poisson integrals

“…Obtained results complement the results of the papers [15]- [16], and also clarify the estimate (5), which was obtained in [9].…”