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 Vallée-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.