# Volume 63, № 6, 2011

### Grüss-type and Ostrowski-type inequalities in approximation theory

Acu A. M., Gonska H., Ra¸sa I.

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 723-740

We discuss the Grass inequalities on spaces of continuous functions defined on a compact metric space. Using the least concave majorant of the modulus of continuity, we obtain a Grass inequality for the functional $L(f) = H(f; x)$, where $H: C[a,b] \rightarrow C[a,b]$ is a positive linear operator and $x \in [a,b]$ is fixed. We apply this inequality in the case of known operators, for example, the Bernstein, Hermite-Fejer operator the interpolation operator, convolution-type operators. Moreover, we derive Grass-type inequalities using Cauchy's mean value theorem, thus generalizing results of Cebysev and Ostrowski. A Grass inequality on a compact metric space for more than two functions is given, and an analogous Ostrowski-type inequality is obtained. The latter in turn leads to one further version of Grass' inequality. In an appendix, we prove a new result concerning the absolute first-order moments of the classical Hermite-Fejer operator.

### On thin-complete ideals of subsets of groups

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 741-754

Let $F \subset \mathcal{P}_G$ be a left-invariant lower family of subsets of a group $G$. A subset $A \subset G$ is called $\mathcal{F}$-thin if
$xA \bigcap yA \in \mathcal{F}$ for any distinct elements $x, y \in G$. The family of all $\mathcal{F}$-thin subsets of G is denoted by $\tau(\mathcal{F})$.
If $\tau(\mathcal{F}) = \mathcal{F}$, then $\mathcal{F}$ is called thin-complete.
The *thin-completion* $\tau*(\mathcal{F})$ of $\mathcal{F}$ is the smallest thin-complete subfamily of $\mathcal{P}_G$ that contains $\mathcal{F}$.
Answering questions of Lutsenko and Protasov, we prove that a set $A \subset G$ belongs to $\tau*(G)$ if and only if for any sequence $(g_n)_{n\in \omega}$ of non-zero elements of G there is $n\in \omega$ such that
$$\bigcap_{i_0,...,i_n \in \{0, 1\}}g_0^{i_0}...g_n^{i_n} A \in \mathcal{F}.$$
Also we prove that for an additive family $\mathcal{F} \subset \mathcal{P}_G$ its thin-completion $\tau*(\mathcal{F})$ is additive. If the group $G$ is countable and torsion-free, then the completion $\tau*(\mathcal{F}_G)$ of the ideal $\mathcal{F}_G$ of finite subsets of $G$ is coanalytic and not Borel in the power-set $\mathcal{P}_G$ endowed with the natural compact metrizable topology.

### Poincare series of the multigraded algebras of *SL *_{2}-invariants

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 755-763

Formulas for computation of the multivariate Poincare series $\mathcal{P}(\mathcal{C}_{d}, z_1, z_2,..., z_n,t)$ and $\mathcal{P}(\mathcal{I}_{d}, z_1, z_2,..., z_n)$, are found, where $\mathcal{C}_{d}, \mathcal{I}_{d}, \;\; {d} = (d_1, d_2,..., d_n)$ are multigraded algebras of joint covariants and joint invariants for n binary forms of degrees $d_1, d_2,..., d_n $.

### Discrete model of the nonsymmetric theory of elasticity

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 764-785

We consider a discrete network of a large number of pin-type homogeneous rods oriented along a given vector and connected by elastic springs at each point. The asymptotic behavior of small oscillations of the discrete system is studied in the case where the distances between the nearest rods tend to zero. For generic non-periodic arrays of rods, we deduce equations describing the homogenized model of the system. It is shown that the homogenized equations correspond to a nonstandard dynamics of an elastic medium. Namely, the homogenized stress tensor in the medium depends linearly not only on the strain tensor but also on the rotation tensor.

### Truncated matrix trigonometric problem of moments: operator approach

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 786-797

We study the truncated matrix trigonometric moment problem. We obtain parametrization of all solutions of this moment problem (in both nondegenerate and degenerate cases) via an operator approach. This parametri-zation establishes a one-to-one correspondence between a certain class of analytic functions and all solutions of the problem. We use important results on generalized resolvents of isometric operators, obtained by M. E. Chumakin.

### Estimates for the best asymmetric approximations of asymmetric classes of functions

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 798-808

Asymptotically sharp estimates are obtained for the best $(\alpha, \beta)$ -approximations of the classes $W^r_{1; \gamma, \delta}$ with natural $r$ by algebraic polynomials in the mean.

### Best bilinear approximations of the classes $S^{\Omega}_{p, \theta}B$ of periodic functions of many variables

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 809-826

We obtain exact-order estimates of the best bilinear approximations of classes $S^{\Omega}_{p, \theta}B$ of periodic functions of many variables in the space $L_q$ for some relations between parameters $p, q, \theta$.

### Eigenvalues and eigenfunctions of the Gellerstedt problem for the multidimensional Lavrent?ev?Bitsadze equation

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 827-832

Eigenvalues and eigenfunctions of the Hellerstedt problems for the Lavrentiev - Bitsadze multidimensional equation are found.

### On the energy and pseudo-angle of Frenet vector fields in $R^n_v$

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 833-839

In this paper, we compute the energy of a Frenet vector field and the pseudo-angle between Frenet vectors for a given non-null curve $C$ in semi-Euclidean space of signature $(n, v)$. It is shown that the energy and pseudo-angle can be expressed in terms of the curvature functions of $C$.

### Rings of almost unit stable rank 1

Vasyunyk I. S., Zabavskii B. V.

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 840-843

We introduce the notion of a ring of almost unit stable rank 1 as generalization of a ring of unit stable rank 1. We prove that the ring of almost unit stable rank 1 with the nonzero Jacobson radical is a ring of unit stable rank 1 and is also a 2-good ring. We introduce the notion of an almost 2-good ring. We show that an adequate domain is an almost 2-good ring.

### Integral representation of even positive-definite functions of two variables

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 844-853

We obtain integral representation of even functions of two variables, for which the kernel $[k_1( x + y) + k_2( x - y)],\quad x, y \in R^2$, is positive definite.

### On fundamental group of Riemannian manifolds with ommited fractal subsets

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 854-858

We show that if $K$ is a closed and bounded subset of a Riemannian manifold $M$ of dimension $m > 3$, and the fractal dimension of $K$ is less than $m - 3$, then the fundamental groups of $M$ and $M - K$ are isomorphic.

### Anatolii Volodymyrovych Skorokhod

Korolyuk V. S., Portenko N. I., Samoilenko A. M.

Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 859 -864