# Volume 55, № 3, 2003

### Algorithms for the Best Simultaneous Uniform Approximation of a Family of Functions Continuous on a Compact Set by a Chebyshev Subspace

Ukr. Mat. Zh. - 2003. - 55, № 3. - pp. 291-306

We generalize the cutting-plane method and the Remez method to the case of the problem of the best simultaneous uniform approximation of a family of functions continuous on a compact set.

### Method of Generalized Moment Representations in the Theory of Rational Approximation (A Survey)

Ukr. Mat. Zh. - 2003. - 55, № 3. - pp. 307-359

We give a survey of the method of generalized moment representations introduced by Dzyadyk in 1981 and its applications to Padé approximations. In particular, some properties of biorthogonal polynomials are investigated and numerous important examples are given. We also consider applications of this method to joint Padé approximations, Padé–Chebyshev approximations, Hermite–Padé approximations, and two-point Padé approximations.

### Statistic $D$-Property of Voronoi Summation Methods of Class $W_{Q^2}$

Ukr. Mat. Zh. - 2003. - 55, № 3. - pp. 360-372

We propose a general method for obtaining Tauberian theorems with remainder for one class of Voronoi summation methods for double sequences of elements of a locally convex, linear topological space. This method is a generalization of the Davydov method of $C$-points.

### On Extendability of Solutions of Differential Equations to a Singular Set

Kaplun Yu. I., Samoilenko V. G.

Ukr. Mat. Zh. - 2003. - 55, № 3. - pp. 373-378

We consider the problem of the extendability of solutions of differential equations to a singular set that consists of points at which the right-hand side of the equation considered is undefined.

### On the Asymptotic Behavior of the Remainder of a Dirichlet Series Absolutely Convergent in a Half-Plane

Mikityuk L. Ya., Sheremeta M. M.

Ukr. Mat. Zh. - 2003. - 55, № 3. - pp. 379-388

For a Dirichlet series \(\sum\nolimits_{n = 1}^\infty {a_n \exp \{ s{\lambda}_n \} } \) with nonnegative exponents and zero abscissa of absolute convergence, we study the asymptotic behavior of the remainder \(\sum\nolimits_{k = n}^\infty {\left| {a_k } \right|\exp \{ {\delta \lambda}_k \} } \) , δ < 0, as *n* → ∞.

### States of Infinite Equilibrium Classical Systems

Ukr. Mat. Zh. - 2003. - 55, № 3. - pp. 389-399

We construct a measure that corresponds to the correlation functions of equilibrium states of infinite systems of classical statistical mechanics. The correlation functions satisfy the Bogolyubov compatibility conditions. We also construct measures that correspond to the correlation functions of nonequilibrium states of infinite systems for the Boltzmann hierarchy and the Bogolyubov–Strel'tsova diffusion hierarchy.

### Multipoint Problem with Multiple Nodes for Partial Differential Equations

Ptashnik B. I., Symotyuk M. M.

Ukr. Mat. Zh. - 2003. - 55, № 3. - pp. 400-413

We establish conditions for the existence and uniqueness of a solution of a problem with multipoint conditions with respect to a selected variable *t* (in the case of multiple nodes) and periodic conditions with respect to *x* _{1},..., *x* _{p} for a nonisotropic partial differential equation with constant complex coefficients. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of this problem.

### Approximation of Continuous Functions by de la Vallée-Poussin Operators

Ukr. Mat. Zh. - 2003. - 55, № 3. - pp. 414-424

For the upper bounds of the deviations of a function defined on the entire real line from the corresponding values of the de la Vallée-Poussin operators, we find asymptotic equalities that give a solution of the well-known Kolmogorov–Nikol'skii problem.

### $l_p$-Solutions of One Difference Equation in a Banach Space

Ukr. Mat. Zh. - 2003. - 55, № 3. - pp. 425-430

We establish a criterion for the existence and uniqueness of solutions of a linear difference equation with an unbounded operator coefficient belonging to the space $l_p(B)$ of sequences of elements of a Banach space $B$.