# Volume 52, № 12, 2000

### On Some Properties of Orthogonal Polynomials over an Area in Domains of the Complex Plane. I

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1587-1595

We establish conditions for the interference of singularities of a weight function and a contour for orthogonal polynomials over the area of a domain. We obtain new estimates for the rate of growth of these polynomials, which depend on the singularities of the weight function and the contour.

### Inequalities of the Jackson Type in the Approximation of Periodic Functions by Fejér, Rogosinski, and Korovkin Polynomials

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1596-1602

We consider inequalities of the Jackson type in the case of approximation of periodic functions by linear means of their Fourier series in the space *L* _{2}. In solving this problem, we choose the integral of the square of the modulus of continuity as a majorant of the square of the deviation. We establish that the constants for the Fejér and Rogosinski polynomials coincide with the constant of the best approximation, whereas the constant for the Korovkin polynomials is greater than the constant of the best approximation.

### Equivalence of Differential Operators in Spaces of Analytic Functions over the Tate Field

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1603-1609

We obtain conditions for the equivalence of certain differential operators in spaces of analytic functions over the Tate field.

### On Bounded Solutions of Some Classes of Two-Parameter Difference Equations in a Banach Space

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1610-1614

We obtain criteria for the existence of bounded solutions of some classes of linear two-parameter difference equations with operator coefficients in a Banach space.

### Asymptotic Discontinuity of Smooth Solutions of Nonlinear $q$-Difference Equations

Derfel' G. A., Romanenko Ye. Yu., Sharkovsky O. M.

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1615-1629

We investigate the asymptotic behavior of solutions of the simplest nonlinear *q*-difference equations having the form *x*(*qt*+ 1) = *f*(*x*(*t*)), *q*> 1, *t*∈ *R* ^{+}. The study is based on a comparison of these equations with the difference equations *x*(*t*+ 1) = *f*(*x*(*t*)), *t*∈ *R* ^{+}. It is shown that, for “not very large” *q*> 1, the solutions of the *q*-difference equation inherit the asymptotic properties of the solutions of the corresponding difference equation; in particular, we obtain an upper bound for the values of the parameter *q*for which smooth bounded solutions that possess the property \(\begin{array}{*{20}c} {\max } \\ {t \in [0,T]} \\ \end{array} \left| {x'(t)} \right| \to \infty \) as *T*→ ∞ and tend to discontinuous upper-semicontinuous functions in the Hausdorff metric for graphs are typical of the *q*-difference equation.

### Optimization Problem on Permutations with Linear-Fractional Objective Function: Properties of the Set of Admissible Solutions

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1630-1640

We consider an optimization problem on permutations with a linear-fractional objective function. We investigate the properties of the domain of admissible solutions of the problem.

### Rings with Elementary Reduction of Matrices

Romaniv A. M., Zabavskii B. V.

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1641-1649

We establish necessary and sufficient conditions under which a quasi-Euclidean ring coincides with a ring with elementary reduction of matrices. We prove that a semilocal Bézout ring is a ring with elementary reduction of matrices and show that a 2-stage Euclidean domain is also a ring with elementary reduction of matrices. We formulate and prove a criterion for the existence of solutions of a matrix equation of a special type and write these solutions in an explicit form.

### Asymptotics of Blaschke Products the Counting Function of Zeros of Which Is Slowly Increasing

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1650-1660

We find the asymptotics as *z*→ 1 for the Blaschke product with positive zeros the counting function of which *n*(*t*) is slowly increasing, i.e., *n*((*t*+ 1)/2) ∼ *n*(*t*) as *t*→ 1.

### $V$-Limit Analysis of Vector-Valued Mappings

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1661-1675

For an arbitrary net of mappings defined on subsets of the Hausdorff space (*X*, τ) and acting into a vector topological space (*Y*, τ) semiordered by a solid cone Λ, we introduce the notion of *V*-limit. We investigate topological and sequential properties of *V*-limit mappings and establish sufficient conditions for their existence. The results presented can be used as a basis for the procedure of averaging of problems of vector optimization.

### On Inequalities of the Landau–Kolmogorov–Hörmander Type on a Segment and Real Straight Line

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1676-1688

We prove inequalities of the Landau–Kolmogorov–Hörmander type for the uniform norms (on some subinterval) of positive and negative parts of intermediate derivatives of functions defined on a finite interval. By using the limit transition, we obtain a new proof or the well-known Hörmander result.

### Approximation of Periodic Analytic Functions by Interpolation Trigonometric Polynomials

Serdyuk A. S., Stepanets O. I.

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1689-1701

We obtain asymptotic equalities for the upper bounds of approximations by interpolation trigonometric polynomials on classes of convolutions of periodic functions admitting a regular extension to a fixed strip of the complex plane.

### Criteria for the Asymptotic Stability of Solutions of Dynamical Systems

Dzhalladova I. A., Valeyev K. G.

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1702-1707

We present a new proof for criteria for the asymptotic stability of systems of difference and differential equations based on the properties of monotone operators in a semiordered space. We also establish necessary and sufficient conditions for the asymptotic stability of stochastic systems of differential and difference equations in the mean square.

### Groups Proper Nonmaximal Subgroups of Which Are Cyclic or Minimal Noncyclic

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1708-1710

We constructively describe locally graded groups all proper nonmaximal subgroups of which are cyclic or minimal noncyclic.

### Joint Continuity and Quasicontinuity of Horizontally Quasicontinuous Mappings

Maslyuchenko V. K., Nesterenko V. V.

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1711-1714

We show that if *X*is a topological space, *Y*satisfies the second axiom of countability, and *Z*is a metrizable space, then, for every mapping *f*: *X*× *Y*→ *Z*that is horizontally quasicontinuous and continuous in the second variable, a set of points *x*∈ *X*such that *f*is continuous at every point from {*x*} × *Y*is residual in *X*. We also generalize a result of Martin concerning the quasicontinuity of separately quasicontinuous mappings.

### On One Counterexample in Convex Approximation

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1715-1721

We prove the existence of a function *f*continuous and convex on [−1, 1] and such that, for any sequence {*p* _{n}}_{ n= 1} ^{∞}of algebraic polynomials *p* _{n}of degree ≤ *n*convex on [−1, 1], the following relation is true: \(\begin{array}{*{20}c} {\lim \sup } \\ {n \to \infty } \\ \end{array} \begin{array}{*{20}c} {\max } \\{x \in [ - 1,1]} \\ \end{array} \frac{{|f(x) - p_n (x)|}}{{\omega _4 (\rho _n (x),f)}} = \infty \) , where ω_{4}(*t*, *f*) is the fourth modulus of continuity of the function *f*and \(\rho _n \left( x \right): = \frac{1}{{n^2 }} + \frac{1}{n}\sqrt {1 - x^2 } \) . We generalize this result to *q*-convex functions.

### Tables of Contents, Volume 52, Numbers 1–12, 2000

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1722-1727