Том 71
№ 1

All Issues

Nesterenko A. N.

Articles: 4
Brief Communications (Ukrainian)

On the Third Moduli of Continuity

Bezkryla S. I., Chaikovs'kyi A. V., Nesterenko A. N.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1420-1424

An inequality for the third uniform moduli of continuity is proved. This inequality implies that an arbitrary 3-majorant is not necessarily a modulus of continuity of order 3.

Article (Ukrainian)

Improvement of one inequality for algebraic polynomials

Chaikovs'kyi A. V., Nesterenko A. N., Tymoshkevych T. D.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 231-242

We prove that the inequality $||g(⋅/n)||_{L_1[−1,1]}||P_{n+k}||_{L_1[−1,1]} ≤ 2||gP_{n+k}||_{L_1[−1,1]}$, where $g : [-1, 1]→ℝ$ is a monotone odd function and $P_{n+k}$ is an algebraic polynomial of degree not higher than $n + k$, is true for all natural $n$ for $k = 0$ and all natural $n ≥ 2$ for $k = 1$. We also propose some other new pairs $(n, k)$ for which this inequality holds. Some conditions on the polynomial $P_{n+k}$ under which this inequality turns into the equality are established. Some generalizations of this inequality are proposed.

Article (Russian)

Multiparameter inverse problem of approximation by functions with given supports

Nesterenko A. N., Radzievskii G. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1116–1127

Let $L_p(S),\;0 < p < +∞$, be a Lebesgue space of measurable functions on $S$ with ordinary quasinorm $∥·∥_p$. For a system of sets $\{B t |t ∈ [0, +∞)^n \}$ and a given function $ψ: [0, +∞) n ↦ [ 0, +∞)$, we establish necessary and sufficient conditions for the existence of a function $f ∈ L_p(S)$ such that $\inf \{∥f − g∥^p_p| g ∈ L_p(S),\;g = 0$ almost everywhere on $S\B t } = ψ (t), t ∈ [0, +∞)^n$. As a consequence, we obtain a generalization and improvement of the Dzhrbashyan theorem on the inverse problem of approximation by functions of the exponential type in $L_2$.

Brief Communications (Russian)

On one problem for comonotone approximation

Nesterenko A. N., Petrova T. O.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1424–1429

For a comonotone approximation, we prove that an analog of the second Jackson inequality with generalized Ditzian - Totik modulus of smoothness $\omega^{\varphi}_{k, r}$ is invalid for $(k, r) = (2, 2)$ even if the constant depends on a function.