2017
Том 69
№ 6

All Issues

Volume 69, № 5, 2017

Anniversaries (Ukrainian)

Oleksandr Ivanovych Stepanets’ (on his 75th birthday)

Romanyuk A. S., Romanyuk V. S., Samoilenko A. M., Savchuk V. V., Serdyuk A. S., Sokolenko I. V.

Full text (.pdf)

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 579

Article (English)

Polynomial inequalities in quasidisks on weighted Bergman space

Abdullayev G. A., Abdullayev F. G., Tunç E.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 582-598

We continue studying on the Nikol’skii and Bernstein –Walsh type estimations for complex algebraic polynomials in the bounded and unbounded quasidisks on the weighted Bergman space.

Article (Russian)

On the moduli of continuity and fractional-order derivatives in the problems of best mean-square approximations by entire functions of the exponential type on the entire real axis

Vakarchuk S. B.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 599-623

The exact Jackson-type inequalities with modules of continuity of a fractional order $\alpha \in (0,\infty )$ are obtained on the classes of functions defined via the derivatives of a fractional order $\alpha \in (0,\infty )$ for the best approximation by entire functions of the exponential type in the space $L_2(R)$. In particular, we prove the inequality $$2^{- \beta /2}\sigma^{- \alpha} (1 - \cos t)^{- \beta /2} \leq \sup \{ \scr {A}_\sigma (f) / \omega_{\beta }(\scr{D}^{\alpha} f, t/\sigma ) : f \in L^{\alpha}_2 (R)\} \leq \sigma^{-\alpha} (1/t^2 + 1/2)^{\beta /2},$$ where $\beta \in [1,\infty ), t \in (0, \pi ], \sigma \in (0,\infty ).$ The exact values of various mean $\nu$ -widths of the classes of functions determined via the fractional modules of continuity and majorant satisfying certain conditions are also determined.

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, $$c(n, k) = \biggl( 1 + k \frac{\sqrt{ 1 + n^2} - 1}{n} \biggr)^2 - k.$$

Article (Ukrainian)

Padé approximants for some classes of multivariate functions

Golub A. P., Lysenko L. O.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 631-640

We extend Dzyadyk’s method of generalized moment representations to the multidimensional case and, on this basis, construct and investigate the Pad´e-type approximants for some classes of multivariate functions.

Article (Ukrainian)

Pointwise estimation of an almost copositive approximation of continuous functions by algebraic polynomials

Dzyubenko H. A.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 641-649

In the case where a function continuous on a segment $f$ changes its sign at $s$ points $y_i : 1 < y_s < y_{s-1} < ... < y_1 < 1$, for any $n \in N$ greater then a constant $N(k, y_i)$ that depends only on $k \in N$ and \$\min_{i=1,...,s-1}\{ y_i - y_{i+1}\}$, we determine an algebraic polynomial $P_n$ of degree \leq n such that: $P_n$ has the same sign as f everywhere except possibly small neighborhoods of the points $y_i$: ($$(y_i \rho_n(y_i), y_i + \rho_n(y_i)),\quad \rho_n(x) := 1/n2 + \sqrt{1 - x^2}/n,$$ $P_n(y_i) = 0$ and $$| f(x) P_n(x)| \leq c(k, s)\omega_k(f, \rho_n(x)),\quad x \in [ 1, 1],$$ where $c(k, s)$ is a constant that depends only on $k$ and $s$ and $\omega k(f, \cdot )$ is the modulus of continuity of the function $f$ of order $k$.

Article (Ukrainian)

Approximating properties of biharmonic Poisson operators in the classes $\hat{L}^{\psi}_{\beta, 1}$

Kharkevych Yu. I., Zhyhallo T. V.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 650-656

We obtain the asymptotic equalities for the least upper bounds of the approximations of functions from the classes $\hat{L}^{\psi}_{\beta, 1}$ by biharmonic Poisson operators in the integral metric.

Article (Ukrainian)

Direct and inverse theorems on the approximation of 2π -periodic functions by Taylor – Abel – Poisson operators

Prestin J., Savchuk V. V., Shydlich A. L.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 657

We obtain direct and inverse theorems on the approximation of 2\pi -periodic functions by Taylor – Abel – Poisson operators in the integral metric.

Article (Russian)

Trigonometric and linear widths for the classes of periodic multivariable functions

Romanyuk A. S.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 670-681

We establish the exact-order estimates for the trigonometric widths of Nikol’skii – Besov $B^r_{\infty ,\theta}$ and Sobolev $W^r_{\infty, \alpha} $ classes of periodic multivariable functions in the space $L_q,\; 1 < q < \infty$. The behavior of the linear widths of Nikol’skii – Besov $B^r_{\infty ,\theta}$ classes in the space $L_q$ is investigated for certain relations between the parameters $p$ and $q$.

Article (Russian)

Kolmogorov widths and entropy numbers in the Orlich spaces with the Luxembourg norm

Romanyuk V. S.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 682-694

We obtain the exact-order estimates of the Kolmogorov widths and entropy numbers of unit balls from the binary Besov spaces \$\mathrm{d}\mathrm{y}\mathrm{a}\mathrm{d} B^{0,\gamma}_{ p,\theta}$ compactly embedded in the exponential Orlich $\mathrm{e}\mathrm{x}\mathrm{p} L^{\nu}$ spaces equipped with the Luxembourg norm.

Article (Ukrainian)

Approximation of the classes of generalized Poisson integrals by Fourier sums in metrics of the spaces $L_s$

Serdyuk A. S., Stepanyuk T. A.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 695-704

In metrics of the spaces $L_s,\; 1 \leq s \leq \infty$, we establish asymptotic equalities for the upper bounds of approximations by Fourier sums in the classes of generalized Poisson integrals of periodic functions that belong to the unit ball of space $L_1$.

Brief Communications (Russian)

On the fractional integrodifferentiation of complex polynomials in $L_0$

Kovalenko L. G., Storozhenko E. A.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 705-710

We establish Bernstein-type inequalities for the fractional integroderivatives of arbitrary algebraic polynomials in the space $L_0$.

Brief Communications (Russian)

Nikol’skii – Stechkin-type inequalities for the increments of trigonometric polynomials in metric spaces

Pichugov S. A.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 711-716

In the spaces $L_{\Psi} [0, 2\pi ]$ with the metric $$\rho (f, 0)\Psi = \frac1{2\pi }\int^{2\pi }_0 \Psi (| f(x)| ) dx,$$ where $\Psi$ is a function of the modulus-ofcontinuity type, we investigate an analog of the Nikol’skii – Stechkin inequalities for the increments and derivatives of trigonometric polynomials.