2018
Том 70
№ 2

# Shevchuk I. A.

Articles: 15
Article (Ukrainian)

### Exact constant in the Dzyadyk inequality for the derivative of an algebraic polynomial

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 624-630

For natural $k$ and $n \geq 2k$, we determine the exact constant $c(n, k)$ in the Dzyadyk inequality $$|| P^{\prime}_n\varphi^{1-k}_n ||_{C[ 1,1]} \leq c(n, k)n\| P_n\varphi^{-k}_n \|_{C[ 1,1]}$$ for the derivative $P^{\prime}_n$ of an algebraic polynomial $P_n$ of degree $\leq n$, where $$\varphi_n(x) := \sqrt{n^{-2} + 1 - x_2,} .$$ Namely, $$c(n, k) = \biggl( 1 + k \frac{\sqrt{ 1 + n^2} - 1}{n} \biggr)^2 - k.$$

Anniversaries (Ukrainian)

### Oleksandr Mykolaiovych Sharkovs’kyi (on his 80th birthday)

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 257-260

Anniversaries (Ukrainian)

### Mykola Oleksiiovych Perestyuk (on his 70th birthday)

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 142-144

Anniversaries (Ukrainian)

### Motornyi Vitalii Pavlovych (on his 75th birthday)

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 995-999

Article (English)

### Shape-preserving projections in low-dimensional settings and the q -monotone case

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 674-684

Let $P: X \rightarrow V$ be a projection from a real Banach space $X$ onto a subspace $V$ and let $S \subset X$. In this setting, one can ask if $S$ is left invariant under $P$, i.e., if $PS \subset S$. If $V$ is finite-dimensional and $S$ is a cone with particular structure, then the occurrence of the imbedding $PS \subset S$ can be characterized through a geometric description. This characterization relies heavily on the structure of $S$, or, more specifically, on the structure of the cone $S^{*}$ dual to $S$. In this paper, шє remove the structural assumptions on $S^{*}$ and characterize the cases where $PS \subset S$. We note that the (so-called) $q$-monotone shape forms a cone which (lacks structure and thus) serves as an application for our characterization.

Article (English)

### Are the degrees of the best (co)convex and unconstrained polynomial approximations the same? II

Ukr. Mat. Zh. - 2010. - 62, № 3. - pp. 369–386

In Part I of the paper, we have proved that, for every $α > 0$ and a continuous function $f$, which is either convex $(s = 0)$ or changes convexity at a finite collection $Y_s = \{y_i\}^s_i = 1$ of points $y_i ∈ (-1, 1)$, $$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(\alpha,s) \sup \left\{n^{\alpha}E_n(f):\; n \geq 1 \right\},$$ where $E_n (f)$ and $E^{(2)}_n (f, Y_s)$ denote, respectively, the degrees of the best unconstrained and (co)convex approximations and $c(α, s)$ is a constant depending only on $α$ and $s$. Moreover, it has been shown that $N^{∗}$ may be chosen to be 1 for $s = 0$ or $s = 1, α ≠ 4$, and that it must depend on $Y_s$ and $α$ for $s = 1, α = 4$ or $s ≥ 2$. In Part II of the paper, we show that a more general inequality $$\sup \left\{n^{\alpha}E^{(2)}_n(f,Y_s):\;n \geq N^{*}\right\} \leq c(\alpha, N, s) \sup \left\{n^{\alpha}E_n(f):\; n \geq N \right\},$$ is valid, where, depending on the triple $(α,N,s)$ the number $N^{∗}$ may depend on $α,N,Y_s$, and $f$ or be independent of these parameters.

Anniversaries (Ukrainian)

### Yuri Yurievich Trokhimchuk (on his 80th birthday)

Ukr. Mat. Zh. - 2008. - 60, № 5. - pp. 701 – 703

Obituaries (Ukrainian)

### Alexander Ivanovich Stepanets

Ukr. Mat. Zh. - 2007. - 59, № 12. - pp. 1722-1724

Chronicles (Ukrainian)

### The international conference „International workshop on analysis and its applications"

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1722

Article (English)

### Coconvex Pointwise Approximation

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1200-1212

Assume that a function fC[−1, 1] changes its convexity at a finite collection Y := {y 1, ... y s} of s points y i ∈ (−1, 1). For each n > N(Y), we construct an algebraic polynomial P n of degree ≤ n that is coconvex with f, i.e., it changes its convexity at the same points y i as f and $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c{\omega }_{2} \left( {f,\frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in \left[ { - 1,1} \right],$$ where c is an absolute constant, ω2(f, t) is the second modulus of smoothness of f, and if s = 1, then N(Y) = 1. We also give some counterexamples showing that this estimate cannot be extended to the case of higher smoothness.

Anniversaries (Ukrainian)

### Oleksandr Ivanovych Stepanets' (on his 60-th birthday)

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 579-580

Brief Communications (Ukrainian)

### Remark on the Lebesgue constant in the Rogosinski Kernel

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 1002–1004

For every n, we compute the Lebesgue constant of Rogosinski kernel with any preassigned accuracy.

Article (Ukrainian)

### Properties of dzyadyk's polynomial kernels

Ukr. Mat. Zh. - 1985. - 37, № 1. - pp. 130 – 132

Article (Ukrainian)

### On a constructive characterization of functions from the classes D r H ω (t) on closed sets with a piecewise smooth boundary

Ukr. Mat. Zh. - 1973. - 25, № 1. - pp. 81—90

Article (Ukrainian)

### On limiting values of an integral of Cauchy type for functions of zygmund classes

Ukr. Mat. Zh. - 1972. - 24, № 5. - pp. 601–617