# Volume 63, № 3, 2011

### On the Bernstein - Walsh-type lemmas in regions of the complex plane

Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 291-302

Let $G \subset C$ be a finite region bounded by a Jordan curve $L := \partial G,\quad \Omega := \text{ext} \; \overline{G}$ (respect to $\overline{C}$), $\Delta := \{z : |z| > 1\}; \quad w = \Phi(z)$ be the univalent conformal mapping of $\Omega$ ont $\Phi$ normalized by $\Phi(\infty) = \infty,\quad \Phi'(\infty) > 0$. Let $A_p(G),\; p > 0$, denote the class of functions $f$ which are analytic in $G$ and satisfy the condition $$||f||^p_{A_p(G)} := \int\int_G |f(z)|^p d \sigma_z < \infty,\quad (∗)$$ where $\sigma$ is a two-dimensional Lebesque measure. Let $P_n(z)$ be arbitrary algebraic polynomial of degree at most $n$. The well-known Bernstein – Walsh lemma says that $$P_n(z)k ≤ |\Phi(z)|^{n+1} ||P_n||_{C(\overline{G})}, \; z \in \Omega. \quad (∗∗)$$ Firstly, we study the estimation problem (∗∗) for the norm (∗). Secondly, we continue studying the estimation (∗∗) when we replace the norm $||P_n||_{C(\overline{G})}$ by $||P_n||_{A_2(G)}$ for some regions of complex plane.

### Intermixing “according to Ibragimov”. Estimate for rate of approach of family of integral functionals of solution of differential equation with periodic coefficients to family of the Wiener processes. Some applications. II

Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 303-318

In the first part of this work, we obtain estimates for the rate of approach of integrals of a family of "physical" white noises to a family of the Wiener processes. By using this result, we establish an estimate for the rate of approach of a family of solutions of ordinary differential equations, disturbed by some physical white noises, to a family of solutions of the corresponding Ito equations. We consider the case where the coefficient of random disturbance is separated from zero as well as the case where it is not separated from zero.

### Rings with finite decomposition of identity

Dokuchaev M. A., Gubareni N. M., Kirichenko V. V.

Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 319-340

A criterion for semiprime rings with finite decomposition of identity to be prime is given. We also give a short survey on some finiteness conditions related to the decomposition of identity. We consider the notion of a net of a ring and show that the lattice of all two-sided ideals of a right semidistributive semiperfect ring is distributive. An application of decompositions of identity to groups of units is given.

### On the theory of convergence and compactness for Beltrami equations

Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 341-340

The convergence and compactness theorems are proved for classes of regular solutions of the Beltrami degenerate equations with restrictions of integral type on the dilatation.

### The rate of pointwise approximation of positive linear operators based on *q*-integer

Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 350-360

This paper is concerned with positive linear operators based on a *q*-integer. The rate of covergence of these operators are established. For these operators, we give Voronovskaya-type theorems and apply them to *q* Bernstein polynomials and *q*-Stancu operators.

### On the injectivity of the Pompeiu transform for integral ball means

Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 361-368

An uniqueness theorem is proved for functions in $\mathbb{R}^n, n \geq 2$, with vanishing integrals over balls of fixed radius and a given majorant of growth. The problem of the unimprovability of this theorem is considered.

### Quasicontinuous approximation in classical statistical mechanics

Petrenko S. M., Rebenko A. L., Tertychnyi M. V.

Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 369-384

A continuous infinite systems of point particles with strong superstable interaction are considered in the framework of classical statistical mechanics. The family of approximated correlation functions is determined in such a way that they take into account only those configurations of particles in the space $\mathbb{R}^d$ which, for a given partition of $\mathbb{R}^d$ into nonintersecting hypercubes with a volume $a^d$, contain no more than one particle in every cube. We prove that so defined approximations of correlation functions pointwise converge to the proper correlation functions of the initial system if the parameter of approximation a tends to zero for any positive values of an inverse temperature $\beta$ and a fugacity $z$. This result is obtained for both two-body and many-body interaction potentials.

### On some properties of generalized quasiisometries with unbounded characteristic

Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 385-398

We consider a family of the open discrete mappings $f:\; D \rightarrow \overline{\mathbb{R}^n}$ that distort in a special way the $p$ -modulus of families of curves connecting the components of spherical condenser in a domain $D$ in $\mathbb{R}^n$, $p > n — 1,\;\; p < n$, and omitting a set of positive $p$-capacity. We establish that this family is normal provided that some function realizing the control of the considered distortion of curve family has a finite mean oscillation at every point or only logarithmic singularities of the order, which is not larger than $n − 1$. We prove that, under these conditions, an isolated singularity $x_0 \in D$ of the mapping $f : D \ \{x_0\} \rightarrow \overline{\mathbb{R}^n}$ is removable and, moreover, the extended mapping is open and discrete. As applications we obtain analogs of the known Liouville and Sokhotski – Weierstrass theorems.

### A generalized mixed type of quartic, cubic, quadratic and additive functional equation

Rassias J. M., Xu T. Z., Xu W. X.

Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 399-415

We determine the general solution of the functional equation $f(x + ky) + f(x — ky) = g(x + y) + g(x — y) + h(x) + \tilde{h}(y)$ forfixed integers $k$ with $k \neq 0, \pm 1$ without assuming any regularity condition on the unknown functions $f, g, h, \tilde{h}$. The method used for solving these functional equations is elementary but exploits an important result due to Hosszii. The solution of this functional equation can also be determined in certain type of groups using two important results due to Szekelyhidi.

### On holomorphic solutions of some boundary-value problems for second-order elliptic operator differential equations

Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 416-420

In the class of holomorphic vector functions, we determine conditions of the solvability of boundary-value problem for a class of second-order differential operator equations, which are given in terms of operator coefficients containing in the equation and in the boundary condition.

### Construction of a solution of one integrodifferential equation

Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 421-426

By using a method proposed by R. Langer, we construct a formal solution of an integral differential equation obtained after the asymptotic integration of one system of linear differential equations with a small parameter of a part of derivatives.

### On the statistical estimation of the initial probability distribution based on the observations of dynamics at the end of an interval

Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 427-431

We consider a problem of the estimation of density of a random value that is an initial value of some dynamics. The dynamics is determined by differential equation whose solution is observable at the end of an interval. This problem is called a problem of the estimation with the use of indirect observations. By using a method of transformation of a measure along an integral curve in combination with kernel estimates, we present a procedure of the estimation of density.