2018
Том 70
№ 9

# Volume 53, № 12, 2001

Article (English)

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

Ukr. Mat. Zh. - 2001. - 53, № 12. - pp. 1588-1599

We study polynomials of higher order orthogonal with weight over domains under the condition that the boundary and weight functions have singularities and do not satisfy the conditions of interference.

Article (Russian)

### On Complementability of Nonmetacyclic Subgroups in Finite $A$-Groups

Ukr. Mat. Zh. - 2001. - 53, № 12. - pp. 1600-1606

We study groups G that satisfy the following conditions: (i) G is a finite solvable group with nonprimary metacyclic second subgroup and (ii) all Sylow subgroups of the group G are elementary Abelian subgroups. We describe the structure of groups of this type with complementable nonmetacyclic subgroups.

Article (Ukrainian)

### Self-Adjoint Operators Generated by Problems of Transmission with Inhomogeneous Conjugation Conditions

Ukr. Mat. Zh. - 2001. - 53, № 12. - pp. 1607-1622

For problems of transmission for higher-order equations with inhomogeneous conjugation conditions, we construct and investigate the corresponding operators in Hilbert spaces. We prove the self-adjointness of these operators for a spectral problem with a parameter in conjugation conditions.

Article (Russian)

### Dirichlet Problem for an Axisymmetric Potential in a Simply Connected Domain of the Meridian Plane

Ukr. Mat. Zh. - 2001. - 53, № 12. - pp. 1623-1641

We develop a method for the reduction of the Dirichlet problem for an axisymmetric potential in a simply connected domain of the meridian plane to a Cauchy singular integral equation. In the case where the boundary of the domain is smooth and satisfies certain additional conditions, we regularize the indicated singular integral equation.

Article (Russian)

### Approximation of $\overline \psi$-Integrals of Periodic Functions by de la Vallée-Poussin Sums (Low Smoothness)

Ukr. Mat. Zh. - 2001. - 53, № 12. - pp. 1641-1653

We investigate the asymptotic behavior of the upper bounds of deviations of linear means of Fourier series from the classes $C_{\infty} ^{\psi}$. In particular, we obtain asymptotic equalities that give a solution of the Kolmogorov – Nikol'skii problem for the de la Vallée-Poussin sums on the classes $C_{\infty} ^{\psi}$.

Article (Ukrainian)

### Approximation of Infinitely Differentiable Periodic Functions by Interpolation Trigonometric Polynomials in Integral Metric

Ukr. Mat. Zh. - 2001. - 53, № 12. - pp. 1654-1663

We obtain asymptotic equalities for the upper bounds of approximations of periodic infinitely differentiable functions by interpolation trigonometric polynomials in the metric of L 1 on the classes of convolutions.

Article (Ukrainian)

### On the Evolution Operator of the Gradient Diffusion Hierarchy for Plane Rotators

Ukr. Mat. Zh. - 2001. - 53, № 12. - pp. 1664-1685

By using a high-temperature cluster expansion, we construct the evolution operator of the BBGKY-type gradient diffusion hierarchy for plane rotators that interact via a summable pair potential in a Banach space containing the Gibbs (stationary) correlation functions. We prove the convergence of this expansion for a sufficiently small time interval. As a result, we prove that weak solutions of the hierarchy exist in the same Banach space. If the initial correlation functions are locally perturbed Gibbs correlation functions, then these solutions are defined on an arbitrary time interval.

Article (Russian)

### Bessel $G$-Functions. I

Ukr. Mat. Zh. - 2001. - 53, № 12. - pp. 1686-1696

We introduce a new class of special functions that generalize the classical Bessel functions and prove their main properties.

Article (Russian)

### Factorization of Periodic Locally Solvable Groups by Locally Nilpotent and Nilpotent Subgroups

Ukr. Mat. Zh. - 2001. - 53, № 12. - pp. 1697-1717

We establish a series of new results concerning periodic locally solvable and finite solvable groups G = AB with locally nilpotent or nilpotent subgroups A and B.

Brief Communications (English)

### Isometric Imbedding of Some Metric Spaces in lp-Spaces

Ukr. Mat. Zh. - 2001. - 53, № 12. - pp. 1718-1723

We present a generalization of a result of Fichet, who proved in 1988 that every ulltrametric space consisting of n points is imbedded isometrically in $l_p^{n - 1} ,p \geqslant 1$ .