# Volume 45, № 6, 1993

### Evolution of the concept of the characteristic function of a linear operator

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 731–743

This is a brief survey of the development and applications of the concept of the characteristic function for different classes of linear operators.

### Commuting extensions of operators

Arlinsky Yu. M., Mogilevskii V. I.

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 744–752

The maximal commuting proper extensions of a closed Hermitian operator and a dual pair of continuous operators in a Hilbert space are described; the criteria of their existence are established.

### On a comparison method for pulse systems in the space $R^n$

Akhmetov M. U., Perestyuk N. A.

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 753–762

A method for the study of differential equations with pulse influence on the surfaces, which was realized in [1] for a bounded domain in the phase space, is now extended to the entire space $R^n$. We prove theorems on the existence of integral surfaces in the critical case and justify the reduction principle for these equations.

### On the curved wedge condition and the continuity moduli of conformal mappings

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 763–769

A Jordan curve $L$ is studied, which satisfies an interior or exterior wedge condition for wedges of various geometric forms. We obtain estimates of the continuity moduli for the conformal mappings of the exterior (interior) of $L$ onto the exterior (interior) of the unit disk.

### Convergence of the series of large-deviation probabilities for sums of independent equally distributed random variables

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 770–784

The series $\sum\nolimits_{n \geqslant 1} {\tau _n P(|S_n | \geqslant \varepsilon n^a )}$ is studied, where $S_n$ are the sums of independent equally distributed random variables, $τ_n$ is a sequence of nonnegative numbers, $α > 0$, and $ɛ > 0$ is an arbitrary positive number. For a broad class of sequences $τ_n$, the necessary and sufficient conditions are established for the convergence of this series for any $ɛ > 0$.

### Uniform estimates for monotone approximation

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 785–790

Uniform estimates are obtained for the monotone approximation of the functions from the generalized Babenko classes.

### Scattering matrix for the wave equation with finite radial potential in the two-dimensional space

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 791–802

Expressions for partial scattering matrices $S_l(\lambda)$ are obtained for all naturall by using Adamyan's result, which establishes a universal relationship between the scattering matrix for the wave equation with finite potential in a even-dimensional space and the characteristic operator function of a special contraction operator, which describes the dissipation of energy from the region of the space containing a scatterer. It is shown that this problem can be reduced to the case of $l = 0$ for all even $l$ and to the case of $l = 1$ for all odd $l$.

### On the uniqueness of the unital divisor of a matrix polynomial over an arbitrary field

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 803–808

Conditions are established under which the unital divisor extracted from a matrix polynomial over an arbitrary field is determined uniquely by its characteristic polynomial. The result obtained is applied to the problem of solving matrix polynomial equations.

### Approximation of cauchy-type integrals in Jordan domains

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 809–833

The concept of a generalized ψ-derivative of a function of a complex variable is introduced and applied to classify functions analytic in Jordan domains. The approximations of functions from the classes introduced by this procedure are studied by using algebraic polynomials constructed on the basis of the Faber polynomials after the summation of Faber series. Analogs of the author's results are obtained for the classes $L_{\beta}^{\psi} \mathfrak{R}$.

### Stability estimates for linear stochastic systems with deviating argument of neutral type

Bychkov A. S., Khusainov D. Ya.

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 834–842

We study linear stochastic differential equations with deviating argument of neutral type and establish sufficient conditions of stability. The functions determining the initial perturbations of solutions are found.

### On a property of the entire dirichlet series with decreasing coefficients

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 843–853

The class $S_{Ψ}^{ *} (A)$ of the entire Dirichlet series $F(s) = \sum\nolimits_{n = 0}^\infty {a_n exp(s\lambda _n )}$ is studied, which is defined for a fixed sequence $A = (a_n ),\; 0 < a_n \downarrow 0,\sum\nolimits_{n = 0}^\infty {a_n< + \infty } ,$ by the conditions $0 ≤ λ_n ↗ +∞$ and $λ_n ≤ (1n^+(1/a_n ))$ imposed on the parameters $λ_n$, where $ψ $ is a positive continuous function on $(0, +∞)$ such that $ψ(x) ↑ +∞$ and $x/ψ(x) ↑ +∞$ as $x →+ ∞$. In this class, the necessary and sufficient conditions are given for the relation $ϕ(\ln M(σ, F)) ∼ ϕ(\ln μ(σ, F))$ to hold as $σ → +∞$, where $M(\sigma ,F) = sup\{ |F(\sigma + it)|:t \in \mathbb{R}\} ,\mu (\sigma ,F) = max\{ a_n exp(\sigma \lambda _n ):n \in \mathbb{Z}_ + \}$, and $ϕ$ is a positive continuous function increasing to $+∞$ on $(0, +∞)$, forwhich $\ln ϕ(x)$ is a concave function and $ϕ(\ln x)$ is a slowly increasing function.

### Calculation of the indicator of an entire function of rational order in terms of its taylor coefficients

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 854–858

By using both the Pólya theorem on the connection between the growth of an entire exponential function and the location of singularities of its Borel transform and the analog of this result for finite-order entire functions (due to Mclntyre), we obtain estimates for the indicator of the growth of an entire function in terms of its Taylor coefficients and, in some cases, determine this indicator exactly.

### On the asymptotic estimates of the best approximations of differentiable functions by algebraić polynomials in the space $L_1$

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 859–862

An asymptotically exact estimate is obtained for the best approximations of $r$ times differentiable functions by algebraic polynomials in the space $L_1$.

### On a transformation of the wiener process in $ℝ^m$ by a functional of the local time type on a surface

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 863–866

A transformation of the Wiener process $ξ_t$ in $ℝ^m$ is considered. This transformation is realized by a multiplicative functional $α_l = u(ξ_l/u(ξ_0)$, where the function $u$ is constructed in a certain way by using a functional of the local time type on a surface. It is proved that this transformation is equivalent to the successive application of an absolutely continuous change of a measure and killing on the surface.

### Asymptotic behavior of the best approximations for the functions in $L_p$

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 867–870

The asymptotic behavior of the ratio $E_n (f)_p/ω(f, α/ n)_p$ is studied for individual periodic functions $f \in L_p$.

### Empirical correlation operator and many-dimensional Hermite polynomials

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 871–875

The action of an empirical correlation operator on the subspaces of vector Hermite polynomials of a given order is studied. The principal part of this operator is selected.

### Topological aspects of dynamical systems on manifolds

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 876–878

Necessary and sufficient conditions are presented for the existence of dynamical systems on manifolds, for which the set of nonwandering points consists of a disconnected union of 2-dimensional tori with a hyperbolic structure.