# Volume 58, № 1, 2006

### Influence of poles on equioscillation in rational approximation

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 3–11

The error curve for the rational best approximation of *ƒ* ? *C*[?1, 1] is characterized by the well-known equioscillation property. Contrary to the polynomial case, the distribution of these alternations is not governed by the equilibrium distribution. It is known that these points need not be dense in [?1, 1]. The reason is the influence of the distribution of the poles of rational approximants. In this paper, we generalize the results known so far to situations where the requirements for the degrees of numerators and denominators are less restrictive.

### Approximation of classes of periodic multivariable functions by linear positive operators

Bushev D. M., Kharkevych Yu. I.

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 12–19

In an *N*-dimensional space, we consider the approximation of classes of translation-invariant periodic functions by a linear operator whose kernel is the product of two kernels one of which is positive. We establish that the least upper bound of this approximation does not exceed the sum of properly chosen least upper bounds in *m*-and ((*N* ? *m*))-dimensional spaces. We also consider the cases where the inequality obtained turns into the equality.

### Structural properties of functions defined on a sphere on the basis of Φ-strong approximation

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 20–25

Structural properties of functions defined on a sphere are determined on the basis of the strong approximation of Fourier-Laplace series.

### Theorems on decomposition of operators in *L*_{1} and their generalization to vector lattices

Maslyuchenko O. V., Mykhailyuk V. V., Popov M. M.

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 26-35

The Rosenthal theorem on the decomposition for operators in *L*_{1} is generalized to vector lattices and to regular operators on vector lattices. The most general version turns out to be relatively simple, but this approach sheds new light on some known facts that are not directly related to the Rosenthal theorem. For example, we establish that the set of narrow operators in *L*_{1} is a projective component, which yields the known fact that a sum of narrow operators in *L*_{1} is a narrow operator. In addition to the Rosenthal theorem, we obtain other decompositions of the space of operators in *L*_{1}, in particular the Liu decomposition.

### Separating functions, spectral theory of graphs, and locally scalar representations in Hilbert spaces

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 36–46

We consider the connection of the separating functions $ρ_r$ with locally scalar representations of graphs and with spectral theory of graphs.

### Problems of approximation theory in linear spaces

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 47–92

We present a survey of results related to the approximation characteristics of the spaces $S^{\rho}_{\varphi}$ and their generalizations. The proposed approach enables one to obtain solutions of problems of classical approximation theory in abstract linear spaces in explicit form. The results obtained yield statements that are new even in the case of approximations in the functional Hilbert spaces $L_2$.

### Linear widths of the classes $B^{\Omega}_{p, \theta}$ of periodic functions of many variables in the space $L_q$

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 93–104

We obtain exact order estimates for the linear widths of the classes $B^{\Omega}_{p, \theta}$ of periodic functions of many variables in the space $L_q$ for certain values of the parameters $p$ and $q$.

### Some properties of a Cauchy-type integral for the Moisil-Theodoresco system of partial differential equations

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 105–112

Our main interest is an analog of a Cauchy-type integral for the theory of the Moisil-Theodoresco system of differential equations in the case of a piecewise-Lyapunov surface of integration. The topics of the paper concern theorems that cover basic properties of this Cauchy-type integral: the Sokhotskii-Plemelj theorem for it as well as a necessary and sufficient condition for the possibility of extending a given Hölder function from such a surface up to a solution of the Moisil-Theodoresco system of partial differential equations in a domain. A formula for the square of a singular Cauchy-type integral is given. The proofs of all these facts are based on intimate relations between the theory of the Moisil-Theodoresco system of partial differential equations and some versions of quaternionic analysis.

### Nikolai Perestyuk (60th birthday)

Mitropolskiy Yu. A., Parasyuk I. O., Samoilenko A. M., ShkiI N. I.

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 113-114

### Some inverse problems for strong parabolic systems

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 115–124

The questions of well-posedness and approximate solution of inverse problems of finding unknown functions on the right-hand side of a system of parabolic equations are investigated. For the problems considered, theorems on the existence, uniqueness, and stability of a solution are proved and examples that show the exactness of the established theorems are given. Moreover, on the set of well-posedness, the rate of convergence of the method of successive approximations suggested for the approximate solution of the given problems is estimated.

### On statistical convergence of vector-valued sequences associated with multiplier sequences

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 125–131

We introduce vector-valued sequence spaces $w_{\infty}(F, Q, p, u), w_{1}(F, Q, p, u), w_{0}(F, Q, p, u), S^q_u$ and $S^q_{0u}$, using a sequence of modulus functions and a multiplier sequence $u = (u_k)$ of nonzero complex numbers. We give some relations for these sequence spaces. It is also shown that if a sequence is strongly $u_q$ -Cesàro summable with respect to the modulus function, then it is $u_q$ -statistically convergent.

### On inverse problem for singular Sturm-Liouville operator from two spectra

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 132–138

In the paper, an inverse problem with two given spectra for second order differential operator with singularity
of type $\cfrac{2}{r} + \cfrac{l(l+1)}{r^2}$
(here, $l$ is a positive integer or zero) at zero point is studied. It is well known that
two spectra $\{\lambda_n\}$ and $\{\mu_n\}$ uniquely determine the potential function $q(r)$ in a singular Sturm-Liouville equation defined on interval $(0, \pi]$.

One of the aims of the paper is to prove the generalized degeneracy of the kernel $K(r, s)$. In particular, we obtain a new proof of Hochstadt's theorem concerning the structure of the difference $\widetilde{q}(r) - q(r)$.

### The space $\Omega^p_m(R^d)$ and some properties

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 139-145

Let $m$ be a $v$-moderate function defined on $R^d$ and let $g \in L^2(R^d)$. In this work, we define $\Omega ^p_m(R^d)$ to be the vector space of $f \in L^2_n(R^d)$ such that the Gabor transform $V_gf$ belongs to $L^p(R^{2d})$, where $1 \leq p < \infty$. We endowe it with a norm and show that it is a Banach space with this norm. We also study some preliminary properties of $\Omega ^p_m(R^d)$. Later we discuss inclusion properties and obtain the dual space of $\Omega ^p_m(R^d)$. At the end of this work, we study multipliers from $L_w^1 (R^d)$ into $\Omega ^p_w(R^d)$ and from $\Omega ^p_w(R^d)$ into $L^{\infty}_{w^{-1}}(R^d)$, where $w$ is Beurling's weight function.

### Еhird summer school algebra, analysis and topology

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 288-289