# Volume 56, № 11, 2004

### Direct and Inverse Problems of Baire Classification of Integrals Depending on a Parameter

Banakh T. O., Kutsak S. M., Maslyuchenko O. V., Maslyuchenko V. K.

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1443-1457

We study the problem of the Baire classification of integrals *g* (*y*) = (*If*)(*y*) = ∫ _{X} *f*(*x, y*)*d*μ(*x*), where *y* is a parameter that belongs to a topological space *Y* and *f* are separately continuous functions or functions similar to them. For a given function *g*, we consider the inverse problem of constructing a function *f* such that *g* = *If*. In particular, for compact spaces *X* and *Y* and a finite Borel measure μ on *X*, we prove the following result: In order that there exist a separately continuous function *f* : *X* × *Y* → ℝ such that *g* = *If*, it is necessary and sufficient that all restrictions *g*|_{ Y } _{ n } of the function *g*: *Y* → ℝ be continuous for some closed covering { *Y* _{ n } *: n* ∈ ℕ} of the space *Y*.

### Best Polynomial Approximations in $L_2$ and Widths of Some Classes of Functions

Shchitov A. N., Vakarchuk S. B.

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1458-1466

We obtain the exact values of extremal characteristics of a special form that connect the best polynomial approximations of functions $f(x) ∈ L_2^r(r ∈ ℤ_{+})$ and expressions containing moduli of continuity of the $k$th order $ω_k(f^{(r)}, t)$. Using these exact values, we generalize the Taikov result for inequalities that connect the best polynomial approximations and moduli of continuity of functions from $L_2$. For the classes $F (k, r, Ψ*)$ defined by $ω_k(f^{(r)}, t)$ and the majorant $Ψ(t)=t^{4k/π^2}$, we determine the exact values of different widths in the space $L_2$.

### Probability Space of Stochastic Fractals

Shpilinskaya O. L., Virchenko Yu. P.

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1467-1484

We develop a general method for the construction of a probability structure on the space F of random sets in ℝ. For this purpose, by using the introduced notion of *c*-system, we prove a theorem on the unique extension of a finite measure from a *c*-system to the minimal *c*-algebra. The obtained structure of measurability enables one to determine probability distributions of the *c*-algebra of random events sufficient, e.g., for the so-called fractal dimensionality of random realizations to be considered as a measurable functional on F.

### Criteria for the Well-Posedness of the Cauchy Problem for Differential Operator Equations of Arbitrary Order

Piven’ A. L., Rutkas A. G., Vlasenko L. A.

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1484-1500

In Banach spaces, we investigate the differential equation \(\mathop \sum \nolimits_{j = 0}^n \;A_j u^{(j)} (t) = 0\) with closed linear operators *A* _{ j } (generally speaking, the operator coefficient *A* _{ n } of the higher derivative is degenerate). We obtain well-posedness conditions that characterize the continuous dependence of solutions and their derivatives on initial data. Abstract results are applied to partial differential equations.

### On the Mean Values of the Dirichlet Series

Sheremeta M. M., Zelisko M. M.

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1501-1502

For Dirichlet series with arbitrary abscissa of absolute convergence, we investigate the relationhip between the increase in the maximum term and \(\left( {\mathop \sum \nolimits_{n = 1}^\infty \left| {a_n } \right|^q \exp \{ q\sigma \lambda _n \} } \right)^{1/q}\) , *q* ∈ (0,+∞).

### Theory of Potential with Respect to Consistent Kernels; Theorem on Completeness and Sequences of Potentials

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1513-1526

The concept of consistent kernels introduced by Fuglede in 1960 is widely used in extremal problems of the theory of potential on classes of positive measures. In the present paper, we show that this concept is also efficient for the investigation of extremal problems on fairly broad classes of signed measures. In particular, for an arbitrary consistent kernel in a locally compact space, we prove a theorem on the strong completeness of fairly general subspaces *E* of all measures with finite energy. (Note that, according to the well-known Cartan counterexample, the entire space *E* is strongly incomplete even in the classical case of the Newton kernel in ℝ^{n} Using this theorem, we obtain new results for the Gauss variational problem, namely, in the non-compact case, we give a description of vague and (or) strong limiting measures of minimizing sequences and obtain sufficient solvability conditions.

### On *C**-Algebras Generated by Deformations of CCR

Kabluchko Z. A., Proskurin D. P., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1527-1538

We consider *C**-algebras generated by deformations of classical commutation relations (CCR), which are generalizations of commutation relations for generalized quons and twisted CCR. We show that the Fock representation is a universal bounded representation. We discuss the connection between the presented deformations and extensions of many-dimensional noncommutative tori.

### Expansion of Weighted Pseudoinverse Matrices in Matrix Power Products

Deineka V. S., Galba E. F., Sergienko I. V.

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1539-1556

On the basis of the Euler identity, we obtain expansions for weighted pseudoinverse matrices with positive-definite weights in infinite matrix power products of two types: with positive and negative exponents. We obtain estimates for the closeness of weighted pseudoinverse matrices and matrices obtained on the basis of a fixed number of factors of matrix power products and terms of matrix power series. We compare the rates of convergence of expansions of weighted pseudoinverse matrices in matrix power series and matrix power products to weighted pseudoinverse matrices. We consider problems of construction and comparison of iterative processes of computation of weighted pseudoinverse matrices on the basis of the obtained expansions of these matrices.

### Best Approximations and Kolmogorov and Trigonometric Widths of the Classes $B_{p,θ}^{Ω}$ of Periodic Functions of Many Variables

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1557-1568

We obtain estimates exact in order for the best approximations and Kolmogorov and trigonometric widths of the classes $B_{p,θ}^{Ω}$ of periodic functions of many variables in the space $L^q$ for certain values of the parameters $p$ and $q$.

### On the Perov Integro-Summable Inequality for Functions of Two Variables

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1569-1576

We present a generalization of the Perov integral inequality for functions of two variables in the case of discontinuous functions.

### On the Relation between Curvature, Diameter, and Volume of a Complete Riemannian Manifold

Nguyen Doan Tuan, Si Duc Quang

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1576–1583

In this note, we prove that if *N* is a compact totally geodesic submanifold of a complete Riemannian manifold *M, g* whose sectional curvature *K* satisfies the relation *K* ≥ *k* > 0, then \(d(m,N) \leqslant \frac{\pi }{{2\sqrt k }}\) for any point *m* ∈ *M*. In the case where dim *M* = 2, the Gaussian curvature *K* satisfies the relation *K* ≥ *k* ≥ 0, and γ is of length *l*, we get Vol (*M, g*) ≤ \(\frac{{2l}}{{\sqrt k }}\) if *k* ≠ 0 and Vol (*M, g* ≤ 2*l*diam (*M*) if *k* = 0.

### The fifteenth scientific session of mathematical commission of the Shevchenko Scientific Society

Pritula N. N., Samoilenko A. M.

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1584