# Volume 62, № 2, 2010

### On the order of relative approximation of classes of differentiable periodic functions by splines

Babenko V. F., Parfinovych N. V.

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 147–157

In the case where $n → ∞$, we obtain order equalities for the best $L_q$ -approximations of the classes $W_p^r ,\; 1 ≤ q ≤ p ≤ 2$, of differentiable periodical functions by splines from these classes.

### On the maximal operator of $(C, α)$-means of Walsh–Kaczmarz–Fourier series

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 158–166

Simon [J. Approxim. Theory, 127, 39–60 (2004)] proved that the maximal operator $σ^{α,κ,*}$ of the $(C, α)$-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space $H_p$ to the space $L_p$ for $p > 1 / (1 + α), \;0 < α ≤ 1$. Recently, Gát and Goginava have proved that this boundedness result does not hold if $p ≤ 1 / (1 + α)$. However, in the endpoint case $p = 1 / (1 + α )$, the maximal operator $σ^{α,κ,*}$ is bounded from the martingale Hardy space $H_{1/(1+α)}$ to the space weak- $L_{1/(1+α)}$. The main aim of this paper is to prove a stronger result, namely, that, for any $0 < p ≤ 1 / (1 + α)$, there exists a martingale $f ∈ H_p$ such that the maximal operator $σ^{α,κ,*} f$ does not belong to the space $L_p$.

### Solvability criterion and representation of solutions of $n$-normal and $d$-normal linear operator equations in a Banach space

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 167–182

On the basis of a generalization of the well-known Schmidt lemma to the case of n-normal and d-normal linear bounded operators in a Banach space, we propose constructions of generalized inverse operators. We obtain criteria for the solvability of linear equations with these operators and formulas for the representation of solutions of these equations.

### On generalization of $⊕$-cofinitely supplemented modules

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 183–189

We study the properties of ⊕-cofinitely radical supplemented modules, or, briefly, $cgs^{⊕}$-modules. It is shown that a module with summand sum property (SSP) is $cgs^{⊕}$ if and only if $M/w \text{Loc}^{⊕} M$ ($w \text{Loc}^{⊕} M$ is the sum of all $w$-local direct summands of a module $M$) does not contain any maximal submodule, that every cofinite direct summand of a UC-extending $cgs^{⊕}$-module is $cgs^{⊕}$, and that, for any ring $R$, every free $R$-module is $cgs^{⊕}$ if and only if $R$ is semiperfect.

### Asymptotic analysis of phase averaging of a transport process

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 190–198

We investigate asymptotic expansions of solutions of singularly perturbed transport equations in Markov and semi-Markov media.

### New equations of infinitesimal deformations of surfaces in $E_3$

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 199–202

We establish a condition for two symmetric tensor fields that is necessary and sufficient for the existence of a displacement vector in the case of infinitesimal deformation of a surface in the Euclidean space E 3.

### On ∗-representations of deformations of canonical anticommutation relations

Proskurin D. P., Sukretnyi K. M.

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 203–214

We consider irreducible ∗-representations of deformations of canonical anticommutation relations (CAR) that belong to the class of ∗-algebras generated by generalized quons.

### On the sets of branch points of mappings more general than quasiregular

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 215–230

It is shown that if a point $x_0 ∊ ℝ^n, \; n ≥ 3$, is an essential isolated singularity of an open discrete $Q$-mapping $f : D → \overline{ℝ^n}, B_f$ is the set of branch points of $f$ in $D$; and a point $z_0 ∊ \overline{ℝ^n}$ is an asymptotic limit of $f$ at the point $x_0$; then, for any neighborhood $U$ containing the point $x_0$; the point $z_0 ∊ \overline{f(B_f ∩ U)}$ provided that the function $Q$ has either a finite mean oscillation at the point $x_0$ or a logarithmic singularity whose order does not exceed $n − 1$: Moreover, for $n ≥ 2$; under the indicated conditions imposed on the function $Q$; every point of the set $\overline{ℝ^n}\ f(D)$ is an asymptotic limit of $f$ at the point $x_0$. For $n ≥ 3$, the following relation is true: $\overline{ℝ^n}∖f(D) ⊂\overline{f(B_f ∩ U)}$. In addition, if $∞ ∉ f(D)$, then the set $f B_f$ is infinite and $x_0 ∈ \overline{B_f}$.

### On closed-form solutions of triple series equations involving Laguerre polynomials

Dhaliwal R.S., Rokne J., Singh B. M.

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 231 – 237

We consider some triple series equations involving generalized Laguerre polynomials. These equations are reduced to triple integral equations for Bessel functions. The closed-form solutions of the triple integral equations for Bessel functions are obtained and, finally, we get the closed-form solutions of triple series equations for Laguerre polynomials.

### Expansion of functions in a system of polynomials biorthogonal on a closed contour with a system of functions regular at infinitely remote point

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 238–254

We study properties of the systems of polynomials constructed according to the schemes similar to the schemes used for the Bernoulli and Euler polynomials, formulate conditions for the existence of functions associated with polynomials and conditions of representation of polynomials by contour integrals, and present the classes of analytic functions expandable in series in the systems of polynomials. The expansions of functions are illustrated by examples.

### Quadruples of orthoprojectors connected by a linear relationship

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 255–264

In the explicit form, we deduce formulas for all quadruples of orthoprojectors $P_1, P_2, P_3$, and $P_4$ irreducible to within unitary equivalence and connected by the linear relationship $α_1 P_1 + α_2 P_2 + α_3 P_3 + α_4 P_4 = λ I$, where $(α_1, α_2, α_3, α_4) ∈ ℝ^{+}$.

### Nonuniqueness of the solution of the gellerstedt space problem for one class of many-dimensional hyperbolic-elliptic equations

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 265–269

It is shown that the solution of the Gellerstedt space problem is not unique for one class of multidimensional hyperbolic-elliptic equations.

### Example of a function of two variables that cannot be an $R$-function

Stegantseva P. G., Velichko I. G.

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 270–274

We note that the definition of R-functions depends on the choice of a certain surjection and pose the problem of the construction of a function of two variables that is not an R-function for any choice of a surjective mapping. It is shown that the function $x_1 x_2 − 1$ possesses this property. We prove a theorem according to which, in the case of finite sets, every mapping is an $R$-mapping for a proper choice of a surjection.

### Block-diagonal reduction of matrices over an $n$-simple Bézout domain $(n ≥ 3)$

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 275–280

It is known that a simple Bézout domain is the domain of elementary divisors if and only if it is 2-simple. The block-diagonal reduction of matrices over an $n$ -simple Bézout domain $(n ≥ 3)$ is realized.

### Integral representation of even positive-definite functions of one variable

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 281 – 284

We obtain an integral representation of even positive-definite functions of one variable for which the kernel $[k_1(x + y) + k_2 (x − y)]$ is positive definite.

### On conditions for the discreteness of the spectrum of a semiinfinite Jacobi matrix with zero diagonal

Khanmamedov A. Kh., Masmaliev G. M.

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 285–288

We establish sufficient conditions for the discreteness of the spectrum of a second-order self-adjoint difference operator generated by a semiinfinite Jacobi matrix with zero principal diagonal.

### Letter to the editor

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 289