# Volume 46, № 11, 1994

### Approximation of continuous vector functions

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1435–1448

We study the possibility of uniform approximation of continuous mappings of metric compact sets into metric spaces. Notions of “weak dimension” and “weak Kolmogorov width” are introduced to compare approximating properties of infinite-dimensional subspaces. For classes of mappings specified by the majorant of the modulus of continuity, we present bilateral estimates of “weak” widths that may coincide under certain conditions.

### Diffusion approximation of normalized integrals of weakly dependent processes and its applications

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1449–1466

We obtain exact estimates for the rate of convergence of normalized integrals of weakly dependent stationary processes to the standard Wiener process in the uniform metric in probability. These estimates are then applied to the investigation of the behavior of stochastic systems with curvilinear boundaries subjected to the action of weakly dependent random perturbations.

### Pointwise estimation of comonotone approximation

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1467–1472

We prove that, for a continuous function *f(x)* defined on the interval [−1,1] and having finitely many intervals where it is either nonincreasing or nondecreasing, one can always find a sequence of polynomials *P* _{ n } *(x)* with the same local properties of monotonicity as the function *f(x)* and such that ¦*f(x)*−*P* _{ n } *(x)* ¦≤*C*ω_{2}(*f*;n^{−2}+*n* ^{−1}√1−*x* ^{2}), where*C* is a constant that depends on the length of the smallest interval.

### Factorization of matrices of permutation type

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1473–1478

By using the methods of the theory of algebraic functions, we present an explicit construction of the canonical factorization of matrices of permutation type given on an open contour.

### On some classes of regular linear extensions of dynamical systems on a torus

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1479–1485

We select some classes of linear extensions of dynamical systems on a torus for which weak regularity implies regularity.

### On sets of regular growth for functions analytic in an open half plane

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1486–1501

We introduce the notion of the set of regular growth for functions analytic in an open half plane. In particular, for a function of completely regular growth in an open half plane, the entire half plane is its set of regular growth. Developed theory is applied to the solution of a problem of Hermitian interpolation in a class of functions of completely regular growth in a half plane with given indicator.

### Asymptotics of a solution of a discontinuous singularly perturbed Cauchy problem

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1502–1508

We construct the asymptotic expansion of a solution of the Cauchy problem for a singularly perturbed system of differential equations whose right-hand side is discontinuous on a certain surface. We consider the case where the surface of discontinuity is crossed and estimate the remainder of the constructed asymptotic expansion.

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

Lopatin A. K., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1509–1526

By using a new method suggested in the first part of the present work, we study systems which become linear in the zero approximation and have perturbations in the form of polynomials. This class of systems has numerous applications. The following fact is even more important: Our technique demonstrates how to generalize the classical method of Poincaré-Birkhoff normal forms and obtain new results by using group-theoretic methods. After a short exposition of the general theory of the method of asymptotic decomposition, we illustrate the new normalization technique as applied to models based on the Lotka-Volterra equations.

### Information complexity of weakly singular integral equations

Makhkamov K. Sh., Pereverzev S. V.

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1527–1533

We establish the exact power order of information complexity for integral equations whose kernels have power singularities and free terms belong to the corresponding Hölder space.

### Optimization of algorithms for the approximate solution of the Volterra equations with infinitely differentiable kernels

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1534–1545

For the Volterra equations with analytic kernels, we establish the exact power order of complexity of their approximate solutions and show that the optimal power order is realized by the method of simple iterations based on the use of information in the form of the values of kernels and free terms at certain points. In addition, for the Volterra equations with infinitely differentiable kernels, we determine the minimal order of the error of direct methods and construct a method which realizes this order.

### On the existence of interpolating $SK$-splines

Serdyuk A. S., Stepanets O. I.

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1546–1553

We establish new sufficient conditions for the existence and uniqueness of generalized interpolating*SK*-splines with uniformly distributed interpolation nodes. Our results include all known important assertions obtained in this field as special cases.

### On the error of the interpolation by bilinear splines

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1554–1560

We study the problem of approximation of functions from the classes W^{r,s} *H* _{ω} and W^{r,s} *H* ^{ω,2}by bilinear splines. For some values of *r* and*s*, we obtain exact estimates of the error.

### On the *N*th diameters of continua

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1561–1563

We study an analog of the problem on the*n* th diameters of continua in the complex plane.

### On the uniqueness of solutions of the Dirichlet and Neumann problems for an elliptic second-order differential equation on a semiaxis

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1564–1567

For an elliptic second-order differential equation in a Banach space, we give a description of all solutions of the homogeneous Dirichlet and Neumann problems and establish conditions under which these problems are uniquely solvable.

### Stochastic analysis and one minimization problem

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1568–1571

We consider the problem of finding the form of a functional of an infinite-dimensional argument for which a certain given expression takes the minimum value for a fixed value of the parameter. The equation obtained for an unknown functional resembles equations with extended stochastic integral.

### Sufficient conditions for the complementability of maximal cyclic subgroups in a finite 2-group

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1572–1575

We present sufficient conditions for each maximal cyclic subgroup of a finite 2-group to be complementable.

### On conditions for convergence of Taylor-Dirichlet series in a convex domain

Krutigolova Ye. K., Mel'nik Yu. I.

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1576-1581

We establish necessary and sufficient conditions for the absolute convergence of the series $$\mathop \sum \limits_{v = 1}^\infty \sum\limits_{k = 0}^{m_v - I} {a_{v,k} z^k \exp (\lambda _v z)} $$ in an open region. We also give conditions under which an arbitrary function analytic in a closed region (analytic in an open region and continuous in a closed region) can be represented by a series of this type.

### On a mixed problem for systems of definite quasilinear partial differential equations with deviating argument

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1581–1585

For a mixed problem for a system of definite quasilinear pseudoparabolic equations with deviating argument, we prove a theorem on differential inequalities and existence of a unique regular solution and a comparison theorem and give sufficient conditions of existence of solutions with constant sign.

### Asymptotic inequalities for the distribution of the time of stay of a semi-Markov process in an expanding set of states

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1586–1590

We establish asymptotic estimates for the behavior of the distribution of the time of the first hit of an infinitely remote level by a semi-Markov process on a semiaxis of integer numbers.

### On the solvability of the Riccati matrix algebraic equation

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1591–1593

We obtain conditions of solvability of the Riccati matrix algebraic equation.

### Two theorems on imbeddings of 0-dimensional groups

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1594-1596

In the class of 0-dimensional groups with infinite weight, the universal group is constructed. We prove that a 0-dimensional group can be imbedded into a multiplicative subgroup of a topological ring.

### Solutions of systems of nonlinear difference-differential equations of neutral type asymptotically bounded in the entire axis

Pelyukh G. P., Samoilenko A. M.

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1597-1601

For a certain class of systems of difference-differential equations of neutral type, we study the properties of solutions asymptotically bounded on the entire axis.

### New condition of harmonicity of functions of infinitely many variables (translation nonpositive case)

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1602-1605

A criterion of harmonicity of functions in a Hilbert space is given in the case of nonnegative second derivatives without using an assumption that they are mutually independent. This assumption is replaced by a weaker condition.