2019
Том 71
№ 2

# Musienko A. P.

Articles: 2
Article (Ukrainian)

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

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 642–653

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.

Article (Ukrainian)

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

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 522-537

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 $C^{ψ}_{β} L_s$.