# Volume 46, № 9, 1994

### Coapproximation of functions convex on a disk by algebraic polynomials

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1115–1122

An estimate is obtained for the approximation of convex functions of two variables with bounded second derivatives on a disk by convex algebraic polynomials.

### On the best polynomial approximation of entire transcendental functions in banach spaces. I

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1123–1133

### On generalized resolvents of Hermitian operators in Krein spaces

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1134–1147

Various classes of extensions and generalized resolvents of Hermitian operators acting in Krein spaces are described in terms of abstract boundary conditions.

### Construction of asymptotic representations of solutions from the classL?? for systems of differential equations by the method of characteristic matrix functions

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1148–1155

### Informativeness of functionals

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1156–1163

We introduce the concept of informativeness of a continuous functional on a metric space*X* with respect to a setM?*X* and a metric ?x. We pose the problem of finding the most informative functional. For some sets of continuous functions, this problem is solved by reduction to a subset of functionals given by the value of a function at a certain point.

### Essential normality of some classes of operators in infinite tensor products of Hilbert spaces

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1164–1170

The essential normality of some classes of operators in stabilized infinite tensor products of Hilbert spaces is established by using the general criteria for the domains of definition of spectral integrals to be essential.

### Bogolyubov averaging and normalization procedures in nonlinear mechanics. I

Lopatin A. K., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1171–1188

We suggest a new method for asymptotic analysis of nonlinear dynamical systems based on group-the-oretic methods. On the basis of the Bogolyubov averaging method, we develop a new normalization procedure — “asymptotic decomposition.” We clarify the contribution of this procedure to the interpretation and development of the averaging method for systems in the standard form and systems with several fast variables. According to this method, the centralized system is regarded as a direct analog of the system averaged in Bogolyubov's sense. The operation of averaging is interpreted as the Bogolyubov projector, i.e., the operation of projection of an operator onto the algebra of centralizer.

### Three-dimensional initial boundary-value problem of the convection of a viscous weakly compressible fluid. II. Uniqueness and stability of generalized solutions

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1189–1202

We continue our study of the three-dimensional initial boundary-value problem of the convection of a viscous thermally inhomogeneous weakly compressible fluid which fills a cavity in a solid body. We prove theorems on the uniqueness of the generalized solution of this problem and its continuity with respect to initial conditions and perturbations. We obtain estimates of exponential type for the decay of solutions (in the mean) for large time.

### Groups satisfying the condition of weak minimality for subgroups of derived length two

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1203–1207

We consider non-Abelian solvable and radical groups that satisfy the condition of weak minimality for subgroups of derived length two. We show that these groups are minimax groups. This is not true for locally solvable groups. We give an example of a group of derived length three that satisfies the condition of weak minimality for the indicated subgroups but is not a minimax group.

### Optimal rates of convergence of some iterative approximation methods for the solution of fredholm equations in spaces of periodic analytic functions

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1208–1215

We consider some classes of Fredholm equations with integral operators acting in spaces of periodic analytic functions. For these classes, we establish the exact order of the optimal rates of convergence for some versions of the method of iterative projections and KP methods.

### Reducibility of nonlinear almost periodic systems of difference equations on an infinite-dimensional torus

Martynyuk D. I., Perestyuk N. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1216–1223

### Approximation of differentiable functions by polynomial splines in domains with external peaks

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1224–1233

### Approximation method for the problems of mechanics of inhomogeneous hereditarily elastic bodies

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1234–1245

We consider a boundary-value problem of mechanics of inhomogeneous hereditarily elastic bodies formulated as a linear equation with an operator of fractional integration, partial derivatives with respect to time and spatial variables, and polynomial-type coefficients of one of the variables. An approximate solution of this problem is constructed according to Dzyadyk's a-method combined with the use of the Laplace transformation. It is proved that the errors of the approximation of the required function and its derivatives decrease in geometric progression.

### Optimization of direct methods for the approximate solution of weakly singular integral equations

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1246–1254

For a class of weakly singular integral equations with power and logarithmic singularities, we establish the optimal order of the error of direct methods and indicate the procedure which realizes this order.

### Some applications of the comparison principle in the theory of perturbed systems

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1255–1260

We establish new estimates for solutions of nonlinear systems.

### Exact constant in the Jackson inequality in the space *L*_{2}

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1261–1265

### On monotone and convex approximation by algebraic polynomials

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1266–1270

### On the rate of convergence of double series of exponents representing regular functions on products of convex polygons

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1271–1274

Estimates exact in order are obtained in the uniform and integral metrics for the deviation of partial sums of double series of exponents that represent functions which are regular on products of convex polygons and either continuous on a product of closed polygons or belonging to the Smirnov class.

### Construction of one-frequency solutions of boundary-value problems for a nonautonomous wave equation

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1275–1279

By using special periodic Ateb-functions, we construct asymptotic representations of one-frequency solutions of boundary-value problems for a nonautonomous wave equation.

### Groups with the minimality condition for infinitely generated subgroups

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1280–1282

We establish that, in the class of locally almost solvable groups, the minimality condition for infinitely generated subgroups is equivalent to the minimality condition for (all) subgroups.

### Dzyadyk formula for summation of double Faber series

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1283–1287