# Volume 57, № 3, 2005

### On Jackson-Type Inequalities for Functions Defined on a Sphere

Babenko V. F., Doronin V. G., Ligun A. A., Shumeiko A. A.

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 291–304

We obtain exact estimates of the approximation in the metrics $C$ and $L_2$ of functions, that are defined on a sphere, by means of linear methods of summation of the Fourier series in spherical harmonics in the case where differential and difference properties of functions are defined in the space $L_2$.

### Polynomial Form of de Branges Conditions for the Denseness of Algebraic Polynomials in the Space $C_w^0$

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 305–319

In the criterion for polynomial denseness in the space $C_w^0$ established by de Brange in 1959, we replace the requirement of the existence of an entire function by an equivalent requirement of the existence of a polynomial sequence. We introduce the notion of strict compactness of polynomial sets and establish sufficient conditions for a polynomial family to possess this property.

### Generalizations of Fox Homotopy Groups, Whitehead Products, and Gottlieb Groups

Golasinski M., Goncalves D., Wong P.

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 320–328

In this paper, we redefine the torus homotopy groups of Fox and give a proof of the split exact sequence of these groups. Evaluation subgroups are defined and are related to the classical Gottlieb subgroups. With our constructions, we recover the Abe groups and prove some results of Gottlieb for the evaluation subgroups of Fox homotopy groups. We further generalize Fox groups and define a group $\tau = \left[ \sum\left(V \times WU*\right), X\right]$ in which the generalized Whitehead product of Arkowitz is again a commutator. Finally, we show that the generalized Gottlieb group lies in the center of $\tau$, thereby improving a result of Varadarajan.

### Asymptotic Equivalence of Triangular Differential Equations in Hilbert Spaces

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 329–337

In this article, we study conditions for the asymptotic equivalence of differential equations in Hilbert spaces. We also discuss the relationship between the properties of solutions of differential equations of triangular form and those of truncated differential equations.

### Asymptotic Behavior of Unbounded Solutions of Essentially Nonlinear Second-Order Differential Equations. I

Evtukhov V. M., Kas'yanova V. A.

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 338–355

We establish asymptotic representations for one class of unbounded solutions of second-order differential equations whose right-hand sides contain a sum of terms with nonlinearities of a more general form than nonlinearities of the Emden-Fowler type.

### On Some Solvable Classes of Nonlinear Nonisospectral Difference Equations

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 356–365

We investigate different measure transformations of the mapping-multiplication type in the cases where the corresponding chains of differential equations can be efficiently found and integrated.

### Solvability and Trajectory-Final Controllability of Pseudohyperbolic Systems

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 366–376

We consider the problem of solvability and optimization for a pseudohyperbolic operator of the general form. We prove theorems on existence and uniqueness for various right-hand sides of the equation. The results obtained are applied to the problem of trajectory-final controllability.

### Boundary-Value Problem for Linear Parabolic Equations with Degeneracies

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 377–387

In spaces of classical functions with power weight, we prove the correct solvability of a boundary-value problem for parabolic equations with an arbitrary power order of degeneracy of coefficients with respect to both time and space variables.

### On the Decomposition of a Diagonal Operator into a Linear Combination of Idempotents or Projectors

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 388–393

For a bounded operator that is not a sum of scalar and compact operators and is similar to a diagonal operator, we prove that it is a linear combination of three idempotents. It is also proved that any self-adjoint diagonal operator is a linear combination of four orthoprojectors with real coefficients.

### Approximation of Continuous Functions of Low Smoothness by de la Vallee-Poussin Operators

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 394–399

We study some problems of the approximation of continuous functions defined on the real axis. As approximating aggregates, the de la Vallee-Poussin operators are used. We establish asymptotic equalities for upper bounds of the deviations of the de la Vallee-Poussin operators from functions of low smoothness belonging to the classes \(\hat C^{\bar \psi } \mathfrak{N}\).

### Weighted Moduli of Smoothness and Sign-Preserving Approximation

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 400–420

We consider a continuous function that changes its sign on an interval finitely many times and pose the problem of the approximation of this function by a polynomial that inherits its sign. For this approximation, we obtain (in the case where this is possible) Jackson-type estimates containing modified weighted moduli of smoothness of the Ditzian-Totik type. In some cases, constants in these estimates depend substantially on the location of points where the function changes its sign. We give examples of functions for which these constants are unimprovable. We also prove theorems that are analogous, in a certain sense, to inverse theorems of approximation without restrictions.

### On Isometric Immersion of Three-Dimensional Geometries $SL_2$, $Nil$ and $Sol$ into a Four-Dimensional Space of Constant Curvature

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 421–426

We prove the nonexistence of isometric immersion of geometries $\text{Nil}^3$, $\widetilde{SL}_2$ into the four-dimensional space $M_c^4$ of the constant curvature $c$. We establish that the geometry $\text{Sol}^3$ cannot be immersed into $M_c^4$ if $c \neq -1$ and find the analytic immersion of this geometry into the hyperbolic space $H^4(-1)$.

### On the Equivalence of Some Conditions for Convex Functions

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 427–431

We study classes of convex functions on $(1, \infty)$ that tend to zero at infinity. Relations between different elements of these classes are determined.