Lebesgue-type inequalities for the de la Vallée-poussin sums on sets of entire functions

Abstract

For functions from the sets C ψ β L s , 1 ≤ s ≤ ∞, where ψ(k) > 0 and $ {\lim_{{k\to \infty }}}\frac{{\psi \left( {k+1} \right)}}{{\psi (k)}} $ , we obtain asymptotically sharp estimates for the norms of deviations of the de la Vallee-Poussin sums in the uniform metric represented in terms of the best approximations of the (ψ, β) -derivatives of functions of this kind by trigonometric polynomials in the metrics of the spaces L s . It is shown that the obtained estimates are sharp on some important functional subsets.