# Volume 51, № 5, 1999

### 50 Years of “Ukrainskii Matematicheskii Zhurnal”

Mitropolskiy Yu. A., Strok V. V.

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 579–580

### Vladislav Kirillovich Dzyadyk (on his 80-th birthday)

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 581-582

### A new method for the construction of solutions of nonlinear wave equations

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 583-593

We propose a simple new method for the construction of solutions of multidimensional nonlinear wave equations.

### On the absolute convergence of power series

Hembars'ka S. B., Zaderei P. V.

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 594–602

We obtain a two-dimensional analog of the Hardy-Littlewood result on the absolute convergence of power series in the case of multiple series on the boundary of a unit polydisk.

### Approximation of fractional-order integrals by algebraic polynomials. I

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 603–613

For functions*f(x)* representable by an integral operator of a special form, we investigate the behavior of the second difference Δ _{ h } ^{2} *f*(*x*)=*f*(*x*+*h*)-2*f*(*x*)+*f*(*x*-*h*),*h*>0, depending on the location of a point*x* on the segment [0,1].

### Stochastic dynamics as a limit of Hamiltonian dynamics of hard spheres

Lampis M., Petrina D. Ya., Petrina К. D.

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 614-635

We consider the stochastic dynamics that is the Boltzmann-Grad limit of the Hamiltonian dynamics of a system of hard spheres. A new concept of averages over states of stochastic systems is introduced, in which the contribution of the hypersurfaces on which stochastic point particles interact is taken into account. We give a rigorous derivation of the infinitesimal operators of the semigroups of evolution operators.

### On the sum of an almost abelian Lie algebra and a Lie algebra finite-dimensional over its center

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 636–644

We consider a Lie algebra*L* over an arbitrary field that is decomposable into the sum*L=A+B* of an almost Abelian subalgebra*A* and a subalgebra*B* finite-dimensional over its center. We prove that this algebra is almost solvable, i.e., it contains a solvable ideal of finite codimension. In particular, the sum of the Abelian and almost Abelian Lie algebras is an almost solvable Lie algebra.

### Weighted approximation in mean of classes of analytic functions by algebraic polynomials and finite-dimensional subspaces

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 645–662

We establish estimates for classic approximation quantities for sets from functional spaces (classes of functions analytic in Jordan domains), namely, for the best polynomial approximations and Kolmogorov widths.

### The theory of the numerical-analytic method: Achievements and new trends of development. V

Ronto M. I., Samoilenko A. M., Trofimchuk S. I.

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 663–673

We analyze the application of the numerical-analytic method proposed by A. M. Samoilenko in 1965 to difference equations.

### Widths and best approximations for classes of convolutions of periodic functions

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 674–687

We establish exact lower bounds for the Kolmogorov widths in the metrics of*C* and*L* for classes of functions with high smoothness; the elements of these classes are sourcewise-representable as convolutions with generating kernels that can increase oscillations. We calculate the exact values of the best approximations of such classes by trigonometric polynomials.

### Several statements for convex functions

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 688–702

For the setM of convex-downward functions Ψ (•) vanishing at infinity, we present its decomposition into subsets with respect to the behavior of special characteristics η (Ψ;•) and μ(Ψ;•) of these functions. We study geometric and analytic properties of the elements of the subsets obtained, which are necessary for the investigation of problems of the theory of approximation for classes of convolutions.

### On the analyticity of the Fourier original and Fourier transform inside alternate angles

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 703–707

We prove that a Fourier image is an analytic function inside two alternate angles if the Fourier preimage possesses the same property.

### On the structure of incoming and outgoing subspaces for a wave equations in $ℝ^n$

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 708-712

We investigate the structure of incoming and outgoing subspaces in the Lax-Phillips scheme for the classic wave equation in $ℝ^n$.

### Projection methods for the solution of Fredholm integral equations of the first kind with $(ϕ, β)$-differentiable kernels and random errors

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 713–717

We estimate errors of projection methods for the solution of the Fredholm equaitons of the first kind*Ax=y+*ζ with random perturbation ζ under the assumption that the integral operator*A* has a (ϕ, β)-differentiable kernel and the mathematical expectation of ∥ξ∥^{2} does not exceed σ^{2}. Under these assumptions, we obtain an estimate that is a complete analog of the well-known result by Vainikko and Plato for the deterministic case where ∥ξ∥≤σ.

### Kato inequality for operators with infinitely many separated variables

Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 718–720

We find conditions under which the Kato inequality is preserved in the case where, instead of an operator with finitely many variables, an operator with infinitely many separated variables is taken. We use the inequality obtained to study both self-adjointness of the perturbed operator with infinitely many separated variables and the domain of definition of the form-sum of this operator and a singular potential.