# Volume 55, № 6, 2003

### Polynomial Asymptotics of Entire Functions of Finite Order

Borova O. I., Zabolotskii N. V.

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 723-732

We obtain new asymptotic relations for entire functions of finite order with zeros on a ray under the condition of regular growth for the counting function of their zeros. These relations improve the well-known results of Valiron.

### Estimates for the Variation of Functions Defined by Double Trigonometric Cosine Series

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 733-749

For functions of two variables defined by trigonometric cosine series with quasiconvex coefficients, we obtain estimates for their variations in the Hardy–Vitali sense.

### Analyticity of Higher-Order Moduli of Continuity of Real-Analytic Functions

Dovgoshei A. A., Potemkina L. L.

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 750-761

The Perel'man's result according to which the first modulus of continuity of any real-analytic function *f* is a function analytic in a certain neighborhood of the origin is generalized to the case of arbitrary moduli of continuity of higher order.

### Markov Games with Several Ergodic Classes

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 762-778

We consider Markov games of the general form characterized by the property that, for all stationary strategies of players, the set of game states is partitioned into several ergodic sets and a transient set, which may vary depending on the strategies of players. As a criterion, we choose the mean payoff of the first player per unit time. It is proved that the general Markov game with a finite set of states and decisions of both players has a value, and both players have ε-optimal stationary strategies. The correctness of this statement is demonstrated on the well-known Blackwell's example (“Big Match”).

### Integral Newton-Type Polynomials with Continual Nodes

Kashpur O. F., Khlobystov V. V., Makarov V. L.

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 779-789

We construct an integral Newton-type interpolation polynomial with a continual set of nodes. This interpolant is unique and preserves an operator polynomial of the corresponding degree.

### Conditions for the Existence of Nonoscillating Solutions of Nonlinear Differential Equations with Delay and Pulse Influence in a Banach Space

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 790-798

For nonlinear differential second-order equations with delay and pulse influence in a Banach space, we establish necessary and sufficient conditions for the existence of their solutions nonoscillating with respect to a subspace.

### Topological Classification of *m*-Fields on Two- and Three-Dimensional Manifolds with Boundary

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 799-805

We consider *m*-fields that are generalizations of the Morse–Smale vector fields for manifolds with boundary. We construct complete topological invariants of *m*-fields on surfaces and *m*-fields without closed trajectories on three-dimensional manifolds. We also prove criteria for the topological equivalence of *m*-fields.

### Approximation of Classes of Analytic Functions by de la Vallée-Poussin Sums

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 806-816

For upper bounds of the deviations of de la Vallée-Poussin sums taken over classes of functions that admit analytic extensions to a fixed strip of the complex plane, we obtain asymptotic equalities. In certain cases, these equalities give a solution of the corresponding Kolmogorov–Nikol'skii problem.

### Descriptive Estimates for a Set of Points That Approximate an Ergodic Measure

↓ Abstract

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 817-823

We obtain descriptive estimates for a set of points that approximate an ergodic invariant measure of a continuous mapping on a compact set. For example, in the case of a metrically transitive mapping with an invariant measure equivalent to the Lebesgue measure, we prove that a set of points generating invariant measures with maximum support contains a dense *G* _{δ}-set, whereas, in the general case, one has a much worse estimate *G* _{δσδ}.

### Convergence of Eigenvalues and Eigenfunctions of Nonlinear Dirichlet Problems in Domains with Fine-Grain Boundary

Namleeva Yu. V., Skrypnik I. V.

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 824-839

We study the behavior of eigenvalues and eigenfunctions of the Dirichlet problem for nonlinear elliptic second-order equations in domains with fine-grain boundary.

### On the Regular Variation of Main Characteristics of an Entire Function

Filevych P. V., Sheremeta M. M.

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 840-849

We establish a necessary and sufficient condition for the coefficients *a* _{n} of an entire function \(f(z) = \sum {_{n = 0}^\infty } {\text{ }}a_n z^n \) under which its central index and the logarithms of the maximum of the modulus and the maximum term are regularly varying functions. We construct an entire function the logarithm of the maximum of whose modulus is a regularly varying function, whereas the Nevanlinna characteristic function is not a regularly varying function.

### On the Spectral Theory of Generalized Toeplitz Kernels

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 850-857

We give a criterion for the denseness of the Fourier transform in the space *L* ^{2} associated with spectral representation of positive-definite Toeplitz kernels.

### Derivation of Moment Equations for Solutions of a System of Nonlinear Difference Equations Dependent on a Semi-Markov Process

Dzhalladova I. A., Valeyev K. G.

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 858-864

We propose a method for the derivation of moment equations for solutions of a system of nonlinear difference equations that depends on a finite-valued semi-Markov process. For systems of linear equations, we compare the results obtained with known ones.