2017
Том 69
№ 9

# Volume 69, № 5, 2017

Anniversaries (Ukrainian)

### Oleksandr Ivanovych Stepanets’ (on his 75th birthday)

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

Article (English)

### Polynomial inequalities in quasidisks on weighted Bergman space

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

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

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

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

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 $\hat{L}^{\psi}_{\beta, 1}$

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

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

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

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 $L_s$

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 $L_0$

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

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.