# Volume 56, № 9, 2004

### On some extremal problems of approximation theory in the complex plane

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1155-1171

In the Hardy Banach spaces *H* _{ q }, Bergman Banach spaces *H*′_{q}, and Banach spaces ℬ (*p, q*, λ), we determine the exact values of the Kolmogorov, Bernstein, Gel’fand, linear, and trigonometric *n*-widths of classes of functions analytic in the disk |z| < 1 and such that the averaged moduli of continuity of their *r*-derivatives are majorized by a certain function. For these classes, we also consider the problems of optimal recovery and coding of functions.

### On properties of Bernstein-typ operator p0lynomials approximating the Urison operator.

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1172-1181

We generalize properties of the Bernstein polynomials to the Bernstein-type operator polynomials approximating the Urison operator.

### Estimation of the best approximation of periodic functions of two variables by an “angle” in the metric of $L_p$

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1182–1192

We obtain upper bounds in terms of Fourier coefficients for the best approximation by an “angle” and for norms in the metric of $L_p$ for functions of two variables defined by trigonometric series with coefficients such that $a_{l_1 l_2} → 0$ as $l_1 + l_2 → ∞$ and $$\mathop \sum \limits_{k_1 = 0}^\infty \mathop \sum \limits_{k_2 = 0}^\infty \left( {\mathop \sum \limits_{l_1 = k_1 }^\infty \mathop \sum \limits_{l_2 = k_2 }^\infty \left| {\Delta ^{12} a_{l_1 \;l_2 } } \right|} \right)^p (k_1 + 1)^{p - 2} \;(k_2 + 1)^{p - 2} < \infty$$ for a certain $p, 1 < p < ∞$.

### Representation of an algebra associated with the Dynkin graph $\tilde E_7$

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1193–1202

We describe the structure of pairs of self-adjoint operators $A$ and $B$ whose spectra belong to the set $\{±1/2, ±3/2\}$ and for which $(A+B)^2 = I$. Such pairs of operators determine a representation of a *-algebra $A_{\tilde E_7 }$ associated with the extended Dynkin graph $\tilde E_7$.

### Truncation method for countable-point boundary-value problems in the space of bounded number sequences

Nedokis V. A., Samoilenko A. M., Teplinsky Yu. V.

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1203-1230

We consider possible methods for the reduction of a countable-point nonlinear boundary-value problem with nonlinear boundary condition on a segment to a finite-dimensional multipoint problem constructed on the basis of the original problem by the truncation method. The results obtained are illustrated by examples.

### Powers of the curvature operator of space forms and geodesics of the tangent bundle

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1231-1243

It is well known that if Г is a geodesic line of the tangent (sphere) bundle with Sasaki metric of a locally symmetric Riemannian manifold, then all geodesic curvatures of the projected curve λ=π _{1463-01} Г are constant. In this paper, we consider the case of the tangent (sphere) bundle over real, complex, and quaternionic space forms and give a unified proof of the following property: All geodesic curvatures of the projected curve are zero beginning with *k* _{3}, *k* _{6}, and *k* _{10} for the real, complex, and quaternionic space forms, respectively.

### Uniform approximation of solutions of nonlinear parabolic problems in perforated domains

Skrypnik I. V., Zhuravskaya A. V.

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1244-1258

We investigate the behavior of a remainder of an asymptotic expansion for solutions of a quasi-linear parabolic Cauchy-Dirichlet problem in a sequence of domains with fine-grained boundary. By using a modification of an asymptotic expansion and new pointwise estimates for a solution of a model problem, we prove the uniform convergence of the remainder to zero.

### On the classification of functional equations on quasigroups

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1259-1266

We consider functional equations over quasigroup operations. We prove that every quadratic parastrophically uncancelable functional equation for four object variables is parastrophically equivalent to the functional equation of mediality or the functional equation of pseudomediality. The set of all solutions of the general functional equation of pseudomediality is found and a criterion for the uncancelability of a quadratic functional equation for four object variables is established.

### Approximation of functions defined on the real axis by operators generated by λ-methods of summation of their Fourier integrals

Kharkevych Yu. I., Zhyhallo T. V.

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1267-1280

We obtain asymptotic equalities for upper bounds of the deviations of operators generated by λ-methods (defined by a collection Λ={λ_{σ}(·)} of functions continuous on [0; ∞) and depending on a real parameter σ) on classes of (ψ, β)-differentiable functions defined on the real axis.

### Separately continuous functions with respect to a variable frame

Herasymchuk V. H., Maslyuchenko O. V., Maslyuchenko V. K.

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1281-1286

We show that the set *D*(*f*) of discontinuity points of a function *f* : **R** ^{2} → **R** continuous at every point *p* with respect to two variable linearly independent directions *e* _{1}(*p*) and *e* _{2}(*p*) is a set of the first category. Furthermore, if *f* is differentiable along one of directions, then *D*(*f*) is a nowhere dense set.

### On the existence of global attractors for one class of cascade systems

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1287-1291

We investigate the qualitative behavior of solutions of cascade systems without uniqueness. We prove that solutions of a reaction-diffusion system perturbed by a system of ordinary differential equations and solutions of a system of equations of a viscous incompressible liquid with passive components form families of many-valued semiprocesses for which a compact global attractor exists in the phase space.

### On the structure of resolvent of singularly perturbed operator solving the problem of eigenvalues

Koshmanenko V. D., Tuhai H. V.

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1292-1297

We investigate the structure of the resolvent of a singularly perturbed operator of finite rank that solves an eigenvalue problem.