# Volume 56, № 1, 2004

### Construction of Floquet–Bloch Solutions and Estimation of Lengths of Resonance Zones of One-Dimensional Schrödinger Equation with Smooth Potential

Denysenko O. M., Parasyuk I. O.

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 3-18

A one-dimensional Schrödinger equation with a potential characterized by a certain rate of approximation by trigonometric polynomials is investigated by methods of the KAM theory. Estimates for resonance energy zones are obtained. The case where the potential belongs to the Gevrey class is analyzed.

### On Stability of Solutions of a Stochastic Equation

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 19-30

We present conditions for the stability of stationary solutions of an abstract linear stochastic differential equation with respect to the coefficient of the leading derivative.

### Application of Potential and Vortex Fields to the Description of Gravitation and Electromagnetism

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 31-50

We propose a system of differential equations in the tensor general-covariant form whose solutions are called gravitational and charged particles. For free fields, solutions are found in the form of Newton and Coulomb potentials.

For a particle that rotates with constant velocity around another particle with large mass, a solution is obtained in the form \(\omega = k\sqrt {\frac{{m_2 }}{{R^3 }}} \) if the particle is uncharged, and in the form \(\omega = k\sqrt {\frac{{\varepsilon _1 \varepsilon _2 }}{{m_1 R^3 }}} \) if it is charged.

### Estimate for the Best Approximation of Summable Functions of Two Variables in Terms of Fourier Coefficients

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 51-69

An upper bound for the best approximation of periodic summable functions of two variables in the metric of *L* is obtained in terms of Fourier coefficients. Functions that can be represented by trigonometric series with coefficients satisfying a two-dimensional analog of the Boas–Telyakovskii conditions are considered.

### On the Approximation by Modified Interpolation Polynomials in Spaces $L_p$

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 70-77

We consider certain modified interpolation polynomials for functions from the space $L_p \;[0, 2π], 1 ≤ p ≤ ∞$. An estimate for the rate of approximation of an original function f by these polynomials in terms of its modulus of continuity is obtained. We establish that these polynomials converge almost everywhere to $f$.

### On the Smoothness of a Generalized Solution of the First Initial Boundary-Value Problem for Strongly Parabolic Systems in a Cylinder with Nonsmooth Base with Respect to Time Variable

Nguyen Manh Hung, Pham Trieu Duong

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 78-87

We consider the first initial boundary-value problem for strongly parabolic systems in an infinite cylinder with nonsmooth boundary. We establish conditions for the existence of generalized solutions, an estimate for this solutions, and an estimate for the derivative of the solution.

### Hopf Algebras and Integrable Flows Related to the Heisenberg–Weil Coalgebra

Blackmore D., Prykarpatsky A. K., Samoilenko A. M.

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 88-96

On the basis of the structure of Casimir elements associated with general Hopf algebras, we construct Liouville–Arnold integrable flows related to naturally induced Poisson structures on an arbitrary coalgebra and their deformations. Some interesting special cases, including coalgebra structures related to the oscillatory Heisenberg–Weil algebra and integrable Hamiltonian systems adjoint to them, are considered.

### Approximation of Poisson Integrals by de la Vallée Poussin Sums

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 97-107

On the classes of Poisson integrals of functions belonging to unit balls in the spaces *C* and *L*, we obtain asymptotic equalities for the upper bounds of approximations by de la Vallée Poussin sums in the uniform metric and the integral metric, respectively.

### Qualitative Investigation of Discontinuous Dynamical Systems on a Plane by the Method of Pointwise Mappings

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 108-118

For two-dimensional discontinuous dynamical systems, we investigate properties of the Poincaré map by the method of pointwise mappings and establish a criterion for the stability of *n*-impulse cycles and an estimate for the number of sinks.

### Convolutions of Singular Distribution Functions

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 119-122

We construct an example that demonstrates that natural restrictions imposed on finite convolutions of singular distributions do not guarantee the purity of the limit distribution.

### Second Jackson Inequality in a Sign-Preserving Approximation of Periodic Functions

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 123-128

We consider a 2π-periodic function *f* continuous on \(\mathbb{R}\)
and changing its sign at 2*s* points *y* _{i} ∈ [−π, π). For this function, we prove the existence of a trigonometric polynomial *T* _{n} of degree ≤*n* that changes its sign at the same points *y* _{i} and is such that the deviation | *f*(*x*) − *T* _{n}(*x*) | satisfies the second Jackson inequality.

### The Best $m$-Term Trigonometric Approximations of the Classes $L_{\beta ,p}^\Psi$ in Uniform Metric

Fedorenko A. S., Fedorenko O.S.

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 129-132

We obtain an exact order estimate for the best approximation of the classes $L_{\beta ,p}^\Psi$ of functions of one variable in the space $L_{∞}$.

### On Saturation of Linear Summation Methods for Fourier Series in the Spaces $S_{\varphi} ^p$

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 133-138

We consider the problem of the saturation of linear summation methods for Fourier series in the spaces $S_{\varphi} ^p,\; р > 0$. We show that the saturation of a linear method and the saturation order are independent of the parameters $X, ϕ$, and p that define the space $S_{\varphi} ^p(X)$.

### Systems of Differential Inequalities with Initial Time Difference

Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 139-145

Some comparison results are formulated for systems of differential inequalities with different initial points.