# Volume 47, № 9, 1995

### On the rate of convergence of methods of projection-iterative type for Fredholm equations with periodic analytic kernels

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1155–1161

Optimal rates of convergence of projection-iterative methods and methods of Sokolov type are found for a certain class of Fredholm equations with analytic kernels that appear within the framework of the method of boundary integral equations.

### On finite $A$-groups with complementable nonmetacyclic subgroups

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1162–1169

We study groups*G* satisfying the following conditions:

(i)*G* is a finite solvable group with non-Abelian commutant and Abelian Sylow subgroups;

(ii) all nonmetacyclic subgroups of*G* are complementable.

We give a description of the structure of such groups with nonmetacyclic second commutant.

### On*n*-widths of bounded periodic holomorphic functions

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1170–1175

The even-dimensional Kolmogorov widths*d* _{ 2n }, Gel'fand widths*d* ^{ 2n }, and linear widths δ_{2n} of*Ã* in*L* _{ q } and*C* are determined exactly. We show that all three*n*-widths are equal and give a characterization of the widths in terms of Blaschke products.

### Canonical spectral equation for empirical covariance matrices

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1176–1189

We study asymptotic properties of normalized spectral functions of empirical covariance matrices in the case of a nonnormal population. It is shown that the Stieltjes transforms of such functions satisfy a socalled canonical spectral equation.

### Approximation of functions of two variables by harmonic splines

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1190-1196

We construct two-dimensional splines and give two versions of an estimate of the deviation of splines from approximated functions. We compare approximations by a planar broken line and by a harmonic spline. We also substantiate the advisability of introduction of the notion of harmonic splines in mathematics.

### Boundary-value problems for differential equations unsolvable with respect to the higher time derivative

Komarnyts'ka L. I., Ptashnik B. I.

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1197–1208

For differential equations with constant coefficients unsolvable with respect to the higher time derivative, we establish conditions of the existence and uniqueness of solutions of problems with conditions local in time and periodic in space variables. We prove a metric theorem on lower bounds of small denominators appearing in the construction of solutions of the problems.

### On construction of asymptotic solution of the perturbed Klein-Gordon equation

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1209-1216

We consider an application of the asymptotic method of nonlinear mechanics to the construction of an approximate solution of the Klein-Gordon equation.

### On quadrature formulas with equal coefficients

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1217–1223

For the Lipschitz classes, we obtain weight quadrature formulas asymptotically optimal with respect to coefficients.

### Diffusion with irregular drift in a Hilbert space

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1224–1230

We construct a generalized diffusion process in a separable Hilbert space. The drift of this process satisfies certain conditions of integrability with respect to the Gaussian measure. We establish the properties of the constructed process.

### On direct methods for solution of regularized equations

Pereverzev S. V., Solodkii S. G.

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1231–1242

We prove that the application of so-called adaptive direct methods to approximation of Fredholm equations of the first kind leads to a more economical way of finite-dimensional approximation as compared with traditional approaches.

### On a method for construction of successive approximations for investigation of multipoint boundary-value problems

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1243–1253

We suggest a new scheme of successive approximations. This scheme allows one to study the problem of existence and approximate construction of solutions of nonlinear ordinary differential equations with multipoint linear boundary conditions. This method enables one to study problems both with singular and nonsingular matrices in boundary conditions.

### A method for solution of nonlinear ordinary differential equations

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1254–1260

We present a method for the solution of nonlinear second-order differential equations by using a system of Fredholm equations of the second kind.

### On the best approximation of classes of convolutions of periodic functions by trigonometric polynomials

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1261–1265

We present sufficient conditions for kernels to belong to the class*N* _{n} ^{*} . In certain cases, this enables us to find exact values of the best approximations of classes of convolutions by trigonometric polynomials in the metrics of*C* and*L*.

### Inverse theorems on approximation of periodic functions

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1266–1273

We obtain new inverse theorems on the approximation of periodic functions $f(·)$ that establish conditions for the existence of their $(ψ, β)$-derivatives. These theorems also guarantee a certain smoothness of these derivatives.

### On uniform approximations of almost periodic functions by entire functions of finite degree

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1274–1279

We give a new proof of the well-known Bernshtein statement that, among entire functions of degree $≤ σ$ which realize the best uniform approximation (of degree $σ$) of a periodic function on $(−∞,∞)$, there is a trigonometric polynomial of degree $≤ σ$. We prove an analog of the mentioned Bernshtein statement and the Jackson theorem for uniform almost periodic functions with arbitrary spectrum.

### On optimal control over quasilinear systems

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1280–1294

We consider the problem of approximately optimal stabilization of quasilinear systems with geometric constraints imposed on control. By using the idea of Krotov global estimates, we justify a method for approximation of the optimal stabilization control and estimate an error in terms of a functional.

### Stability of differential systems with rational right-hand sides

Khusainov D. Ya., Shevelo V. N.

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1295–1299

We prove theorems which establish estimates for the domain of stability of a differential system with rational right-hand side. We also construct estimates of the convergence of solutions of the system.

### An approach to the investigation of optimal quadrature formulas for singular integrals with fixed singularity

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1300–1304

For classes of functions given on the segment [0,1], we obtain optimal quadrature formulas for singular integrals with fixed singularity. The obtained results are extended to the case of two-dimensional integrals.