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We obtain asymptotic formula for upper bounds of deviations of Fejer sums on classes of Poisson integrals. Under certain conditions, formula guarantee the solvability of the Kolmogorov–Nikol’skiy problem for Fejer sums and classes of Poisson integrals.

We study polynomial solutions of gyrostat motion equations under potential and gyroscopic forces applied and of gyrostat motion equations in magnetic field taking into account Barnett–London effect. Mathematically, either of the above mentioned problems is described by a system of non-linear ordinary differential equations whose right hand sides contain fifteen constant parameters. These parameters characterize the gyrostat mass distribution, as well as potential and non-potential forces acting on gyrostat. We consider polynomial solutions of Steklov–Kovalevski–Gorjachev and Doshkevich classes. The structure of invariant relations for polynomial solutions shows that, as a rule, on top of the fifteen parameters mentioned one should add no less than twenty five problem parameters. In the process of solving such a multi-parametric problem in this paper we (in addition to analytic approach) apply numeric methods based on CAS. We break our studies of polynomial solutions existence into two steps. During the first step, we estimate maximal degrees of polynomials considered and obtain a non-linear algebraic system for parameters of differential equations and polynomial solutions. In the second step (using the above CAS software) we study the solvability conditions of the system obtained and investigate the conditions of the constructed solutions to be real.

We construct two new polynomial solutions for Kirchhoff–Poisson. The first one is described by the following property: the projection squares of angular velocity on the non-baracentric axes are the fifth degree polynomials of the angular velocity vector component of the baracentric axis that is represented via hypereliptic function of time. The second solution is characterized by the following: the first component of velocity conditions is a second degree polynomial, the second component is a polynomial of the third degree, and the square of the third component is the sixth degree polynomial of the auxiliary variable that is an inversion of the elliptic Legendre integral.

The third new partial solution we construct for gyrostat motion equations in the magnetic field with Barnett–London effect. Its structure is the following: the first and the second components of the angular velocity vector are the second degree polynomials, and the square of the third component is a fourth degree polynomial of the auxiliary variable which is found via inversion of the elliptic Legendre integral of the third kind.

All the solutions constructed in this paper are new and do not have analogues in the fixed point dynamics of a rigid body.

The paper deals with the problems of approximation of periodic functions of high smoothness by arithmetic means of Fourier sums. The simplest and natural example of a linear process of approximation of continuous periodic functions of a real variable is the approximation of these functions by partial sums of the Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the entire class of continuous $2\pi$-periodic functions. In connection with this, a significant number of papers is devoted to the study of the approximative properties of other approximation methods, which are generated by certain transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for each function $f \in C$. In particular, over the past decades, de la Vallee Poussin sums and Fejer sums have been widely studied. One of the most important directions in this field is the study of the asymptotic behavior of upper bounds of deviations of arithmetic means of Fourier sums on different classes of periodic functions. Methods of investigation of integral representations of deviations of polynomials on the classes of periodic differentiable functions of real variable originated and received its development through the works of S.M. Nikol’sky, S.B. Stechkin, N.P. Korneichuk, V.K. Dzadyk, etc.

The aim of the work systematizes known results related to the approximation of classes of periodic functions of high smoothness by arithmetic means of Fourier sums, and presents new facts obtained for particular cases. In the paper is studied the approximative properties of $r$-repeated de la Vallee Poussin sums on the classes of periodic functions that can be regularly extended into the fixed strip of the complex plane. We obtain asymptotic formulas for upper bounds of the deviations of repeated de la Vallee Poussin sums taken over classes of periodic analytic functions. In certain cases, these formulas give a solution of the corresponding Kolmogorov–Nikolsky problem. We indicate conditions under which the repeated de la Vallee Poussin sums guarantee a better order of approximation than ordinary de la Vallee Poussin sums.

The different types of the reduced equations are known in the dynamics a heavy rigid body with a fixed point. Since the Euler−Poisson’s equations admit the three first integrals, then for the first approach the obtaining new forms of equations are usually based on these integrals. The system of six scalar equations can be transformed to a third-order system with them. However, in indicated approach the reduced system will have a feature as in the form of radical expressions a relatively the components of the angular velocity vector. This fact prevents the effective the effective application of numerical and asymptotic methods of solutions research. In the second approach the different types of variables in a problem are used: Euler’s angles, Hamilton’s variables and other variables. In this approach the Euler−Poisson’s equations are reduced to either the system of second-order differential equations, or the system for which the special methods are effective. In the article the method of finding the reduced system based on the introduction of an auxiliary variable is applied. This variable characterizes the mixed product of the angular momentum vector, the vector of vertical and the unit vector barycentric axis of the body. The system of four differential equations, two of which are linear differential equations was obtained. This system has no analog and does not contain the features that allows to apply to it the analytical and numerical methods. Received form of equations is applied for the analysis of a special class of solutions in the case when the center of mass of the body belongs to the barycentric axis. The variant in which the sum of the squares of the two components of the angular momentum vector with respect to not barycentric axes is constant. It is proved that this variant exists only in the Steklov’s solution. The obtained form of Euler−Poisson’s equations can be used to the investigation of the conditions of existence of other classes of solutions. Certain perspectives obtained equations consists a record of all solutions for which the center of mass is on barycentric axis in the variables of this article. It allows to carry out a classification solutions of Euler−Poisson’s equations depending on the order of invariant relations. Since the equations system specified in the article has no singularities, it can be considered in computer modeling using numerical methods.