# Volume 55, № 9, 2003

### On One Criterion for the Holomorphy of an Arbitrary Mapping of a Plane Domain into a Plane

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1155-1166

We prove the holomorphy of a function that, at every point, preserves either angles or dilations with respect to a certain set.

### Jackson-Type Inequalities in the Space $S^p$

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1167-1177

In the case of approximation of periodic functions in the space $S^p$, we determine the exact constants in Jackson-type inequalities for the Zygmund, Rogosinski, and de la Valleé Poussin linear summation methods.

### Equilibrium Potentials with External Fields

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1178-1195

We investigate the Gauss variational problem over fairly general classes of Radon measures in a locally compact space **X**. We describe potentials of minimizing measures, establish their characteristic properties, and prove the continuity of extremals. Extremal problems dual to the original one are formulated and solved. The results obtained are new even in the case of classical kernels and the Euclidean space \(\mathbb{R}^n \) .

### Almost-Everywhere Convergence and (*o*)-Convergence in Rings of Measurable Operators Associated with a Finite von Neumann Algebra

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1196-1205

We study the relationship between (*o*)-convergence and almost-everywhere convergence in the Hermite part of the ring of unbounded measurable operators associated with a finite von Neumann algebra. In particular, we prove a theorem according to which (*o*)-convergence and almost-everywhere convergence are equivalent if and only if the von Neumann algebra is of the type *I*.

### Singular Integral Operators in Spaces of Oscillating Functions on a Rectifiable Curve

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1206-1217

We prove generalized Noether theorems for a singular integral equation with Cauchy kernel on a closed rectifiable Jordan curve in classes of piecewise-continuous functions with oscillation-type discontinuities. We obtain results concerning the normal solvability of operators associated with the equation and acting into a Banach space and incomplete normed spaces of piecewise-continuous oscillating functions.

### On One Method for Factorization of Algebraic Polynomials

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1218-1223

We propose a method for the factorization of algebraic polynomials with real or complex coefficients and construct a numerical algorithm, which, along with the factorization of a polynomial with multiple roots, solves the problem of the determination of multiplicities and the number of multiple roots of the polynomial.

### On Homomorphisms of Algebras Generated by Projectors and Coxeter Functors

Popovich S. V., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1224-1237

We consider algebras generated by idempotents in Banach spaces and orthoprojectors in Hilbert spaces whose sum is a multiple of the identity. We construct several functors generated by homomorphisms of the algebras considered between categories of representations. We investigate properties of these functors and present their applications.

### Group-Theoretic Description of Riemannian Spaces

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1238-1248

We show that the geometric structure of an arbitrarily curved Riemannian space is locally determined by a deformed group of its diffeomorphisms.

### Induced Representations of Abelian Groups of Finite Rank

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1249-1253

We prove that any irreducible faithful representation of an almost torsion-free Abelian group *G* of finite rank over a finitely generated field of characteristic zero is induced from an irreducible representation of a finitely generated subgroup of the group *G*.

### On Zeros of One Class of Functions Analytic in a Half-Plane

Sharan V.L., Vynnyts’kyi B. V.

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1254-1259

We describe sequences of zeros of functions *f* ≢ 0 analytic in the half-plane \({\mathbb{C}}_ + = \{ z:\operatorname{Re} z >0\}\) and satisfying the condition \((\exists {\tau}_1 \in (0;1))(\exists c_1 >0)(\forall z \in {\mathbb{C}}_ + ):|f(z)| \leqslant c_1 \exp ({\eta}^{\tau }_1 (c_1 |z|)),\) where η: [0; +∞) → (0; +∞) is an increasing function such that the function ln η(*r*) is convex with respect to ln *r* on [1; +∞).

### Distribution of a Random Continued Fraction with Markov Elements

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1260-1268

We study the structure of the distribution of a random variable such that the elements of its continued fraction form a Markov chain of order *m*. It is proved that an absolutely continuous component is absent from such distributions.

### On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank

Dudkin M. Ye., Koshmanenko V. D.

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1269-1276

We discuss purely singular finite-rank perturbations of a self-adjoint operator *A* in a Hilbert space ℋ. The perturbed operators \(\tilde A\) are defined by the Krein resolvent formula \((\tilde A - z)^{ - 1} = (A - z)^{ - 1} + B_z \) , Im *z* ≠ 0, where *B* _{z} are finite-rank operators such that dom *B* _{z} ∩ dom *A* = |0}. For an arbitrary system of orthonormal vectors \(\{ \psi _i \} _{i = 1}^{n < \infty } \) satisfying the condition span |ψ_{ i }} ∩ dom *A* = |0} and an arbitrary collection of real numbers \({\lambda}_i \in {\mathbb{R}}^1\) , we construct an operator \(\tilde A\) that solves the eigenvalue problem \(\tilde A\psi _i = {\lambda}_i {\psi}_i , i = 1, \ldots ,n\) . We prove the uniqueness of \(\tilde A\) under the condition that rank *B* _{z} = *n*.

### The Case Where the Sum of Three Partial Reflections is Equal to Zero

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1277-1283

Up to unitary equivalence, we describe all irreducible triples of self-adjoint operators *A* _{1}, *A* _{2}, *A* _{3} such that σ(*A* _{i}) ⊂ |−1, 0, 1}, *i* = 1, 2, 3, and *A* _{1} + *A* _{2} + *A* _{3} = 0.

### Algebras of Functionally Invariant Solutions of the Three-Dimensional Laplace Equation

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1284-1290

In commutative associative third-rank algebras with principal identity over a complex field, we select bases such that hypercomplex monogenic functions constructed in these bases have components satisfying the three-dimensional Laplace equation. The notion of monogeneity for these functions is similar to the notion of monogeneity in the complex plane.

### On Multidimensional Generalized Diffusion Processes

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1291-1296

We construct a multidimensional generalized diffusion process with the drift coefficient that is the (generalized) derivative of a vector-valued measure satisfying an analog of the Hölder condition with respect to volume. We prove the existence and continuity of the density of transition probability of this process and obtain standard estimates for this density. We also prove that the trajectories of the process are solutions of a stochastic differential equation.