# Volume 71, № 3, 2019

### On the law of the iterated logarithm for the maximum scheme in Banach ideal spaces

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 3. - pp. 303-309

Asymptotic estimates are obtained in the law of the iterated logarithm for the extreme values of a sequence of independent random variables in Banach spaces.

### On one inequality for the moduli of continuity of fractional order generated by semigroups of operators

Bezkryla S. I., Chaikovs'kyi A. V., Nesterenko A. N.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 3. - pp. 310-324

A new inequality for the moduli of continuity of fractional order generated by semigroups of operators is obtained. This inequality implies a generalization of the well-known statement that there exists an $α$-majorant, which is not a modulus of continuity of order $α$ generated by a semigroup of operators, to the case of noninteger values of $α$.

### Multidimensional associated fractions with independent variables and multiple power series

Bodnar D. I., Dmytryshyn R. I.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 3. - pp. 325-349

We establish the conditions of existence and uniqueness of a multidimensional associated fraction with independent variables corresponding to a given formal multiple power series and deduce explicit relations for the coefficients of this fraction. The relationship between the multidimensional associated fraction and the multidimensional $J$ -fraction with independent variables is demonstrated. The convergence of the multidimensional associated fraction with independent variables is investigated in some domains of the space $C^N$. The expansions of some functions into the corresponding two-dimensional associated fraction with independent variables are constructed and the efficiency of approaching of the obtained expansions by approximants is shown.

### Strong summability of two-dimensional Vilenkin – Fourier series

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 3. - pp. 340-352

We study the exponential uniform strong summability of two-dimensional Vilenkin – Fourier series. In particular, it is proved that the two-dimensional Vilenkin – Fourier series of a continuous function $f$ is uniformly strongly summable to a function $f$ exponentially in the power 1/2. Moreover, it is proved that this result is best possible.

### Almost coconvex approximation of continuous periodic functions

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 3. - pp. 353-367

If a $2\pi$ -periodic function $f$ continuous on the real axis changes its convexity at $2s, s \in N$, points $y_i : \pi \leq y_{2s} < y_{2s-1} < . . . < y_1 < \pi$ , and, for all other $i \in Z$, $y_i$ are periodically defined, then, for every natural $n \geq N_{y_i}}$, we determine a trigonometric polynomial $P_n$ of order cn such that $P_n$ has the same convexity as $f$ everywhere except, possibly, small neighborhoods of the points $y_i : (y_i \p_i /n, y_i + \pi /n)$, and $\| f P_n\| \leq c(s) \omega 4(f, \pi /n)$,, where $N_{y_i}}$ is a constant depending only on $\mathrm{m}\mathrm{i}\mathrm{n}_{i = 1,...,2s}\{ y_i y_{i+1}\} , c$ and $c(s)$ are constants depending only on $s, \omega 4(f, \cdot )$ is the fourth modulus of smoothness of the function $f$, and $\| \cdot \|$ is the max-norm.

### The Bojanov – Naidenov problem for functions with nonsymmetric restrictions on the highest derivative

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 3. - pp. 368-381

For given $r \in \bfN , p, \alpha , \beta , \mu > 0$, we solve the extreme problems $$\int^b_ax^q_{\pm} (t)dt \rightarrow \mathrm{s}\mathrm{u}\mathrm{p}, q \geq p,$$ in the set of pairs $(x, I)$ of functions $x \in L^r_{\infty}$ and intervals $I = [a, b] \subset R$ satisfying the inequalities $\beta \leq x(r)(t) \leq \alpha$ for almost all $t \in R$ , the conditions $L(x_{\pm})p \leq L\bigl(( \varphi^{\alpha ,\beta}_{\lambda ,r}) \bigr)_p$, and the corresponding condition $\mu\Bigl(\mathrm{s}\mathrm{u}\mathrm{p} \mathrm{p}_{[a,b]}x_{+}\Bigr) \leq \mu$ or $\mu \Bigl( \mathrm{s}\mathrm{u}\mathrm{p} \mathrm{p}_{[a,b]}x \Bigr) \leq \mu$, where $$L(x)p := \mathrm{s}\mathrm{u}\mathrm{p} \Bigl\{ \| x\| L_{p[a,b]} : a, b \in R , | x(t)| > 0, t \in (a, b)\Bigr\},$$ $\mathrm{s}\mathrm{u}\mathrm{p} \mathrm{p}_{[a,b]}x_{\pm} := \{ t \in [a, b] : x_{\pm} (t) > 0\} , \varphi^{\alpha ,\beta}_{\lambda ,r}$ is the nonsymmetric $(2\pi /\lambda)$-periodic Euler spline of order $r$. As a consequence, we solve the same problems for the intermediate derivatives $x(k)_{\pm} , k = 1,..., r_1,$ with $q \geq 1$.

### Geometric properties of metric spaces

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 3. - pp. 382-399

We study some problems of geometrization of arbitrary metric spaces. In particular, we studied the concept of straight and flat placement of points in this space. In a certain way, we continue the investigations of Kagan devoted to the detailed analysis of the notion of straightforwardness based on four groups of postulates. The results of our work are based on the notion of angular characteristics of three points of the space proposed by Alexandrov. We establish the conditions under which the set of points of an arbitrary metric space satisfies all five postulates of the first group of Kagan’s placement postulates. The relationship between rectilinear and flat placements of points of the metric space is investigated. Examples of placements of this kind based on linear functions in some classical spaces are presented. The results of the paper are obtained without using the property of completeness of the space and can be used for the discrete computation and structuring of specific metric spaces.

### $\scr{Z^{ \ast}}$ - semilocal modules and the proper class $\scr{RS}$

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 3. - pp. 400-411

Over an arbitrary ring, a module $M$ is said to be $\scr{Z^{ \ast}}$ -semilocal if every submodule $U$ of $M$ has a $\scr{Z^{ \ast}}$ -supplement $V$ in $M$, i.e., $M = U + V$ and $U \cap V \subseteq \scr{Z^{ \ast}} (V )$, where $\scr{Z^{ \ast}}(V ) = \{m \in V | Rm$ is a small module $\}$ is the $\mathrm{R}\mathrm{a}\mathrm{d}$-small submodule. In this paper, we study basic properties of these modules as a proper generalization of semilocal modules. In particular, we show that the class of $\scr{Z^{ \ast}}$ -semilocal modules is closed under submodules, direct sums, and factor modules. Moreover, we prove that a ring $R$ is $\scr{Z^{ \ast}}$ -semilocal if and only if every injective left R-module is semilocal. In addition, we show that the class $\scr{RS}$ of all short exact sequences $E :0 \xrightarrow{\psi} M \xrightarrow{\phi} K \rightarrow 0$ such that $\mathrm{I}\mathrm{m}(\psi )$ has a $\scr{Z^{ \ast}}$ -supplement in $N$ is a proper class over left hereditary rings. We also study some homological objects of the proper class $\scr{RS}$ .

### Unicity theorems with truncated multiplicities of meromorphic mappings in several complex variables for few fixed targets

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 3. - pp. 412-432

The purpose of our paper is twofold. Our first aim is to prove a uniqueness theorem for meromorphic mappings of $C^n$ into $P^N(C)$ sharing $2N + 2$ hyperplanes in the general position with truncated multiplicities, where all common zeros with multiplicities more than a certain number do not need to be counted. Second, we consider the case of mappings sharing less than $2N + 2$ hyperplanes. These results are improvements of some recent results.

### Differential operators specifying the solution of an elliptictype iterated equation

Aleksandrovich I. N., Sidorov M. V.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 3. - pp. 433-440

We construct differential operators that transform arbitrary holomorphic functions into regular solutions of elliptic-type equations of the second and higher orders. The Riquier problem is solved for the elliptic-type equation of the fourth order.