Estimates for the Approximations of the Classes of Analytic Functions by Interpolation Analogs of the De-La-Vallée–Poussin Sums

Abstract

We establish two-sided estimates for the exact upper bounds of approximations by the interpolation analogs of the de-la-Vallée-Poussin sums on the classes of 2π -periodic functions C _β,s ^ψ specified by the sequences ψ(k) and shifts of the argument β , β ∈ ℝ, under the condition that the sequences ψ(k) satisfy the d’Alembert D _q , q ∈ (0, 1), condition. Similar estimates are obtained for the classes C _β ^ψ H _ω generated by convex moduli of continuity ω(t). Under the conditions n − p → ∞ and p → ∞, the indicated estimates turn into asymptotic equalities.