# Volume 60, № 8, 2008

### On the best polynomial approximation of entire transcendental functions of generalized order

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1011–1026

We prove a Hadamard-type theorem which connects the generalized order of growth $\rho^*_f(\alpha, \beta)$ of entire transcendental function $f$ with coefficients of its expansion into the Faber series. The theorem is an original extension of a certain result by S. K. Balashov to the case of finite simply connected domain $G$ with the boundary $\gamma$ belonging to the S. Ya. Al'per class $\Lambda^*.$
This enables us to obtain boundary equalities that connect $\rho^*_f(\alpha, \beta)$ with the sequence of the best polynomial approximations of $f$ in some Banach spaces of functions analytic in $G$.

### Problem of impulsive regulator for one dynamical system of the Sobolev type

Rutkas A. G., Samoilenko A. M., Vlasenko L. A.

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1027–1034

We establish conditions for the existence of an optimal impulsive control for an implicit operator differential equation with quadratic cost functional. The results obtained are applied to the filtration problem.

### On maximal stable orders on an inverse semigroup of finite rank with zero

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1035–1041

We consider maximal stable orders on semigroups that belong to a certain class of inverse semigroups of finite rank.

### Lower bound for the best approximations of periodic summable functions of two variables and their conjugates in terms of Fourier coefficients

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1042–1050

In terms of Fourier coefficients, we establish lower bounds for the sum of norms and the sum of the best approximations by trigonometric polynomials for functions from the space *L*(*Q*²) and functions conjugate to them with respect to each variable and with respect to both variables, provided that these functions are summable.

### Asymptotics of approximation of ψ-differentiable functions of many variables

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1051–1057

We investigate approximative characteristics of classes of ψ-differentiable multivariable functions introduced by A. I. Stepanets. We give asymptotics of the approximation of functions from these classes.

### Cone inequalities and stability of differential systems

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1058–1074

We investigate generalizations of classes of monotone dynamical systems in a partially ordered Banach space. We establish algebraic conditions for the stability of equilibrium states of differential systems on the basis of linearization and application of derivatives of nonlinear operators with respect to a cone. Conditions for the positivity and absolute stability of a certain class of differential systems with delay are proposed. Several illustrative examples are given.

### Rate of convergence of the price of European option on a market for which the jump of stock price is uniformly distributed over an interval

Mishura Yu. S., Soloveiko O. M.

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1075–1086

We consider a model of the market such that a jump of share price is uniformly distributed on some symmetric interval and establish the rate of convergence of fair prices of European options by using the theorem on asymptotic decompositions of distribution function for the sum of independent identically distributed random variables. We show that, in the prelimit model, there exists a martingale measure on the market such that the rate of convergence of prices of European options to the Black - Scholes price is of order 1/*n* ^{1/2}.

### On the problem of approximation of functions by algebraic polynomials with regard for the location of a point on a segment

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1087–1098

We obtain a correction of an estimate of the approximation of functions from the class *W ^{ r }H^{ ω}* (here, ω(

*t*) is a convex modulus of continuity such that

*t*ω '(

*t*) does not decrease) by algebraic polynomials with regard for the location of a point on an interval.

### Problem of optimal control for a determinate equation with interaction

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1099–1109

The problem of optimal control of differential equations with interaction is consider. It is proved that the optimal control satisfies the maximum principle and there exists the generalized optimal control. It is shown that, in the considered problem, new technical aspects arise as compared with the usual problem of optimal control.

### Characterizations of the Shunkov groups

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1110–1118

The structure of the family of finite subgroups of the form *L _{g } * = ‹

*a, a*› in periodic Shunkov's group is studied. As a colorraries of the result obtained, two characterizations of periodic Shunkov's groups follow.

^{g }### On the best *L*_{2 }-approximations of functions by using wavelets

Babenko V. F., Zhiganova G. S.

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1119 – 1127

We obtain the exact Jackson-type inequalities for approximations in *L*_{2 }(*R*) of functions *f*∈ *L*_{2 }(*R*)
with the use of partial sums of the wavelet series in the case of the Meyer wavelets and the Shannon–Kotelnikov wavelets.

### Interval distribution function of a bounded chaotic sequence as a basis of nonaxiomatic probability theory

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1128–1137

We introduce the notion of interval distribution function of random events on the set of elementary events and the notion of interval function of the frequencies of these events. In the limiting case, the interval function turns into the ordinary distribution function and the interval function of frequencies (under certain conditions) turns into the density of distribution of random events. The case of discrete sets of elementary events is also covered. This enables one to introduce the notion of the probability of occurrence of random events as a result of the limit transition.

### Solutions of the Kirkwood-Salsburg equation for particles with finite-range nonpairwise repulsion

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1138–1143

For a system of classical particles interacting via stable pairwise integrable and positive many-body (nonpairwise) finite-range potentials, we prove the existence of a solution of the symmetrized Kirkwood-Salsburg equation.

### Domain of convergence of the Euler transform for the power series of an analytic function

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1144–1152

We consider the Euler transform of the power series of an analytic function playing the role of its expansion in a series in a system of polynomials and study the domain of convergence of the transform depending on the parameter of transformation and the character of singular points of the function. It is shown that the transform extends the function beyond the boundaries of the disk of convergence of its series on the interval of the boundary located between two singular points of the function. In particular, it is established that the power series of the function whose singular points are located on a single ray is summed by the transformation in the half plane.