# Volume 64, № 5, 2012

### Oleksandr Ivanovych Stepanets’ (on the 70 th anniversary of his birthday)

Gorbachuk M. L., Lukovsky I. O., Makarov V. L., Motornyi V. P., Romanyuk A. S., Samoilenko A. M., Serdyuk A. S., Sharko V. V., Zaderei P. V.

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 579-581

### On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles

Abdullayev F. G., Özkartepe N. P.

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 582-596

Let $\mathbb{C}$ be the complex plane, let $\overline{\mathbb{C}} = \mathbb{C} \bigcup \{\infty\}$, let $G \subset \mathbb{C}$ be a finite Jordan domain with $0 \in G$, let $L := \partial G$, let $\Omega := \overline{\mathbb{C}} \ \overline{G}$, and let $w = \varphi(z)$ be the conformal mapping of $G$ onto a disk $B(0, \rho) := \{w : \; |w | < \rho_0\}$ normalized by $\varphi(0) = = 0,\; \varphi'(0) = 1$, where $\rho_0 = \rho_0 (0, G)$ is the conformal radius of $G$ with respect to 0. Let $\varphi \rho(z) := \int^z_0 [\varphi'(\zeta)]^{2/p}d\zeta$ and let $\pi_{n,p}(z)$ be the generalized Bieberbach polynomial of degree $n$ for the pair $(G, 0)$ that minimizes the integral $\int\int_G|\varphi'(z) - P'_n(z)|^p d \sigma_z$ in the class of all polynomials of degree $\text{deg} P_n \leq n$ such that $P_n(0) = 0$ and $P'_n(0) = 1$. We study the uniform convergence of the generalized Bieberbach polynomials $\pi_{n,p}(z)$ to $\varphi \rho(z)$ on $\overline{G}$ with interior and exterior zero angles determined depending on properties of boundary arcs and the degree of their tangency. In particular, for Bieberbach polynomials, we obtain better estimates for the rate of convergence in these domains.

### On the dependence of the norm of a function on the norms of its derivatives of orders $k$ , $r - 2$ and $r , 0 < k < r - 2$

Babenko V. F., Kovalenko O. V.

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 597-603

We establish conditions for a system of positive numbers $M_{k_1}, M_{k_2}, M_{k_3}, M_{k_4}, \; 0 = k_1 < k2 < k3 = r − 2, k4 = r$, necessary and sufficient for the existence of a function $x \in L^r_{\infty, \infty}(R)$ such that $||x^{(k_i)} ||_{\infty} = M_{k_i},\quad i = 1, 2, 3, 4$.

### Best mean-square approximation of functions defined on the real axis by entire functions of exponential type

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 604-615

Exact constants in Jackson-type inequalities are calculated in the space $L_2 (\mathbb{R})$ in the case where the quantity of the best approximation $\mathcal{A}_{\sigma}(f)$ is estimated from above by the averaged smoothness characteristic $\Phi_2(f, t) = \cfrac 1t \int^t_0||\Delta^2_h(f)||dh$. We also calculate the exact values of the average $\nu$-widths of classes of functions defined by $\Phi_2$.

### Fourier cosine and sine transforms and generalized Lipschitz classes in uniform metric

Golubov B. I., Volosivets S. S.

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 616-627

For functions $f \in L^1(\mathbb{R}_{+})$ with cosine (sine) Fourier transforms $\widehat{f}_c(\widehat{f}_s)$ in $L^1(\mathbb{R})$, we give necessary and sufficient conditions in terms of $\widehat{f}_c(\widehat{f}_s)$ for $f$ to belong to generalized Lipschitz classes $H^{\omega, m}$ and $h^{\omega, m}$. Conditions for the uniform convergence of the Fourier integral and for the existence of the Schwartz derivative are also obtained.

### Reverse inequalities for geometric and power means

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 628-635

We establish exact bounds for the positive and negative exponents of summability of the power mean of a function in the case where this mean satisfies the reverse Jensen inequality.

### Inequalities for derivatives of functions on an axis with nonsymmetrically bounded higher derivatives

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 636-648

For nonperiodic functions $x \in L^r_{\infty}(\textbf{R})$ defined on the entire real axis, we prove analogs of the Babenko inequality. The obtained inequalities estimate the norms of derivatives $||x^{(k)}_{\pm}||_{L_q[a, b]}$ on an arbitrary interval $[a,b] \subset R$ such that $x^{(k)}(a) = x^{(k)}(b) = 0$ via local $L_p$-norms of the functions $x$ and uniform nonsymmetric norms of the higher derivatives $x(r)$ of these functions.

### Strong summability and properties of Fourier?Laplace series on a sphere

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 649-661

We investigate the behavior of quantities that characterize the strong summability of Fourier - Laplace series. On this basis, we establish some properties of the Fourier - Laplace series of functions of the class $L_2(S^{m-1})$.

### Lower bounds for the deviations of the best linear methods of approximation of continuous functions by trigonometric polynomials

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 662-673

In the case of uniform approximation of continuous periodic functions of one variable by trigonometric polynomials, we obtain lower bounds for the Jackson constants of the best linear methods of approximation.

### 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.

### Best bilinear approximations of functions from Nikolskii-Besov classes

Romanyuk A. S., Romanyuk V. S.

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 685-697

We obtain exact-order estimates for the best bilinear approximations of Nikol'skii-Besov classes in the spaces of functions $L_q (\pi_{2d})$.

### Approximation by interpolation trigonometric polynomials on classes of periodic analytic functions

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 698-712

We establish asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities on the sets $C^{\psi}_{\beta}L_p$ of $(\psi, \beta)$-differentiable functions generated by sequences $\psi(k)$ that satisfy the d'Alembert conditions. We find asymptotic equalities for the least upper bounds of approximations by interpolation trigonometric polynomials on the classes $C^{\psi}_{\beta, p},\;\; 1 \leq p \leq \infty$.

### On the properties of blocks of terms of the series $\sum \cfrac1k \sin kx$

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 713-718

We investigate the decomposability of the series $\sum \cfrac1k \sin kx$ into blocks such that the sum of the series formed of the moduli of these blocks belongs to the spaces $L^p[0, \pi]$ or the spaces $L^p[0, \pi]$ with weight $x^{-\gamma},\quad \gamma < 1$.

### On the best approximations of functions defined on zero-dimensional groups

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 719-728

We present a survey of results of the author, his postgraduates, and other mathematicians related to the problem of finding the best approximations of functions in the investigation of properties of spaces of functions defined on zero-dimensional compact commutative groups.