2018
Том 70
№ 6

All Issues

Shevchuk I. A.

Articles: 17
Article (English)

On moduli of smoothness with Jacobi weights

Kopotun K. A., Leviatan D., Shevchuk I. A.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 379-403

We introduce the moduli of smoothness with Jacobi weights $(1 x)\alpha (1+x)\beta$ for functions in the Jacobi weighted spaces $L_p[ 1, 1],\; 0 < p \leq \infty $. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted spaces $L_p$. If $1 \leq p \leq \infty$, then these moduli are equivalent to certain weighted $K$-functionals (and so they are equivalent to certain weighted Ditzian – Totik moduli of smoothness for these $p$), while for $0 < p < 1$ they are equivalent to certain “Realization functionals”.

Article (Ukrainian)

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

Galan V. D., Shevchuk I. A.

↓ Abstract

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)

Fedorenko V. V., Ivanov А. F., Khusainov D. Ya., Kolyada S. F., Maistrenko Yu. L., Parasyuk I. O., Pelyukh G. P., Romanenko O. Yu., Samoilenko V. G., Shevchuk I. A., Sivak A. G., Tkachenko V. I., Trofimchuk S. I.

Full text (.pdf)

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

Anniversaries (Ukrainian)

Mykola Oleksiiovych Perestyuk (on his 70th birthday)

Boichuk A. A., Gorbachuk M. L., Gorodnii M. F., Khruslov E. Ya., Lukovsky I. O., Makarov V. L., Parasyuk I. O., Samoilenko A. M., Samoilenko V. G., Sharkovsky O. M., Shevchuk I. A., Slyusarchuk V. Yu., Stanzhitskii A. N.

Full text (.pdf)

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

Anniversaries (Ukrainian)

Motornyi Vitalii Pavlovych (on his 75th birthday)

Babenko V. F., Davydov O. V., Kofanov V. A., Parfinovych N. V., Pas'ko A. N., Romanyuk A. S., Ruban V. I., Samoilenko A. M., Shevchuk I. A., Shumeiko A. A., Timan M. P., Trigub R. M., Vakarchuk S. B., Velikin V. L.

Full text (.pdf)

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

Article (English)

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

Prophet M. P., Shevchuk I. A.

↓ Abstract   |   Full text (.pdf)

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

Kopotun K. A., Leviatan D., Shevchuk I. A.

↓ Abstract   |   Full text (.pdf)

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)

Berezansky Yu. M., Bojarski B., Gorbachuk M. L., Kopilov A. P., Korolyuk V. S., Lukovsky I. O., Mitropolskiy Yu. A., Portenko N. I., Reshetnyak Yu. G., Samoilenko A. M., Sharko V. V., Shevchuk I. A., Skorokhod A. V., Tamrazov P. M., Zelinskii Yu. B.

Full text (.pdf)

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

Obituaries (Ukrainian)

Alexander Ivanovich Stepanets

Gorbachuk M. L., Lukovsky I. O., Mitropolskiy Yu. A., Romanyuk A. S., Rukasov V. I., Samoilenko A. M., Serdyuk A. S., Shevchuk I. A., Zaderei P. V.

Full text (.pdf)

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

Chronicles (Ukrainian)

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

Samoilenko A. M., Shevchuk I. A., Stepanets O. I.

Full text (.pdf)

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

Article (English)

Coconvex Pointwise Approximation

Dzyubenko H. A., Gilewicz J., Shevchuk I. A.

↓ Abstract   |   Full text (.pdf)

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)

Lukovsky I. O., Makarov V. L., Mitropolskiy Yu. A., Romanyuk A. S., Romanyuk V. S., Rukasov V. I., Samoilenko A. M., Serdyuk A. S., Shevchuk I. A., Zaderei P. V.

Full text (.pdf)

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

Article (Russian)

A brief survey of scientific results of E. A. Storozhenko

Kashin B. S., Korneichuk N. P., Shevchuk I. A., Ul'yanov P. L.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 463-473

We present a survey of the scientific results obtained by E. A. Storozhenko and related results of her disciples and give brief information about the seminar on the theory of functions held under her guidance.

Brief Communications (Ukrainian)

Remark on the Lebesgue constant in the Rogosinski Kernel

Dzyadyk V. K., Shevchuk I. A.

↓ Abstract   |   Full text (.pdf)

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

Shevchuk I. A.

Full text (.pdf)

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

Shevchuk I. A.

Full text (.pdf)

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

Dzyadyk V. K., Shevchuk I. A.

Full text (.pdf)

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