# Volume 61, № 2, 2009

### Approximation of unbounded operators by bounded operators in a Hilbert space

Babenko V. F., Bilichenko R. O.

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 147-153

We determine the best approximation of an arbitrary power $A^k$ of an unbounded self-adjoint operator $A$ in a Hilbert space $H$ on the class $\{x ∈ D(A^r ) : ∥A^r x∥ ≤ 1\},\; k < r$.

### Serial rings and tiled orders of width two

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 154-159

We construct Artinian serial rings and tiled orders of width two with maximal finite global dimension.

### Integral representations of generalized axially symmetric potentials in a simply connected domain

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 160-177

We obtain integral representations of generalized axially symmetric potentials via analytic functions of a complex variable that are defined in an arbitrary simply connected bounded domain symmetric with respect to the real axis. We prove that these integral representations establish a one-to-one correspondence between analytic functions of a complex variable that take real values on the real axis and generalized axially symmetric potentials of certain classes.

### State estimation for a dynamical system described by a linear equation with unknown parameters

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 178-194

We investigate the state estimation problem for a dynamical system described by a linear operator equation with unknown parameters in a Hilbert space. In the case of quadratic restrictions on the unknown parameters, we propose formulas for *a priori* mean-square minimax estimators and *a posteriori* linear minimax estimators. A criterion for the finiteness of the minimax error is formulated. As an example, the main results are applied to a system of linear algebraic-differential equations with constant coefficients.

### Mixed problem for the Petrovskii well-posed equation in a cylindrical domain

Guseinova É. S., Iskenderov B. A.

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 214-230

We study the problem of existence and uniqueness of the solution of a mixed problem for the Petrovskii well-posed equation in a cylindrical domain and the behavior of this solution for large values of time.

### Improvement of one inequality for algebraic polynomials

Chaikovs'kyi A. V., Nesterenko A. N., Tymoshkevych T. D.

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 231-242

We prove that the inequality $||g(⋅/n)||_{L_1[−1,1]}||P_{n+k}||_{L_1[−1,1]} ≤ 2||gP_{n+k}||_{L_1[−1,1]}$, where $g : [-1, 1]→ℝ$ is a monotone odd function and $P_{n+k}$ is an algebraic polynomial of degree not higher than $n + k$, is true for all natural $n$ for $k = 0$ and all natural $n ≥ 2$ for $k = 1$. We also propose some other new pairs $(n, k)$ for which this inequality holds. Some conditions on the polynomial $P_{n+k}$ under which this inequality turns into the equality are established. Some generalizations of this inequality are proposed.

### Construction of continuous cocycles for the bicrossed product of locally compact groups

Chapovsky Yu., Podkolzin G. B.

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 243-260

For locally compact groups $K, M$, and $N$ such that $M$ and $N$ are subgroups of $K, K = M ∙ N$ and $M ∩ N = \{e\}$, where $e$ is the identity of the group $K$, we give a complete description and propose a method for the construction of pairs of continuous cocycles used in the structure of bicrossed product with cocycles in terms of continuous 2-cocycles on the groups $M, N$, and $$K and 3-cocycles on the group $K$.

### On the existence of a generalized asymmetric (α, β)-spline whose average values have equal minima at given points

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 261-267

We solve the problem of existence of an asymmetric spline averaged in Steklov’s sense that takes equal minimum values at given points.

### Conditions for the existence and uniqueness of bounded solutions of nonlinear differential equations

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 268-279

We establish conditions required for the existence and uniqueness of bounded solutions of the nonlinear differential equation $f_1\left(\frac{dx(t)}{dt} \right) = f_2(x(t)), t ∈ ℝ$.

### Solution of a countable system of quasilinear partial differential equations multiperiodic in a part of variables

Berzhanov A. B., Kurmangaliev E. K.

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 280-288

We establish sufficient conditions for the existence and uniqueness of solutions of a countable system of first-order quasilinear partial differential equations multiperiodic in a part of variables.