2019
Том 71
№ 5

# Lebesgue-type inequalities for the de la Valee-Poussin sums on sets of analytic functions

Abstract

For functions from the sets $C^{ψ}_{β} C$ and $C^{ψ}_{β} L_s,\; 1 ≤ s ≤ ∞$ generated by sequences $ψ(k) > 0$ satisfying the d’Alembert condition $\lim_{k→∞}\frac{ψ(k + 1)}{ψ(k)} = q,\; q ∈ (0, 1)$, we obtain asymptotically unimprovable estimates for the deviations of de la Vallee Poussin sums in the uniform metric in terms of the best approximations of the $(ψ, β)$-derivatives of functions of this sort by trigonometric polynomials in the metrics of the spaces $L_s$. It is proved that the obtained estimates are unimprovable in some important functional subsets of $C^{ψ}_{β} C$ and $C^{ψ}_{β} L_s$.

English version (Springer): Ukrainian Mathematical Journal 65 (2013), no. 4, pp 575-592.

Citation Example: Musienko A. P., Serdyuk A. S. Lebesgue-type inequalities for the de la Valee-Poussin sums on sets of analytic functions // Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 522-537.

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