# Mathematics

## Recurrence Relations for Polynomials Orthonormal on Sobolev, Generated by Laguerre Polynomials

In this paper we consider the system of polynomials (l_r,n)^a (r — natural number, n = 0,1,...), orthonormal with respect to the Sobolev inner product (Sobolev orthonormal polynomials) of the following type<f, g> = (sum _(v=0))^(r−1) f^(ν)(0)g ^(ν)(0) + (f _0)^∞ f^(r) (x)g^(r)(x)ρ^(x)dx and generated by the classical orthonormal Laguerre polynomials.Recurrence relations are obtained for the system of Sobolev orthonormal polynomials, which can be used for studying various properties of these polynomials and calculate their values for any x and n.

## Approximation Properties of Dicrete Fourier Sums for Some Piecewise Linear Functions

Let N be a natural number greater than 1. We select N uniformly distributed points t_k = 2πk/N (0 < k < N − 1) on [0,2\pi]. Denote by L_ n,N (f) = L _n,N (f,x)1 < n < ⌊N/2⌋ the trigonometric polynomial of order n

## Almost Periodic at Infinity Functions Relative to the Subspace of Functions Integrally Decrease at Infinity

In the paper we introduce and study a new class of almost periodic at infinity functions, which is defined by means of a subspace of integrally decreasing at infinity functions. It

is wider than the class of almost periodic at infinity functions introduced in the papers of A.G.Baskakov (with respect to the subspace of functions vanishing at infinity). It suffices to turn

to the approximation theory for a new class of functions, where the Fourier coefficients are slowly varying at infinity functions with respect to the subspace of functions that decrease integrally

## Adjustment of Functions and Lagrange Interpolation Based on the Nodes Close to the Legendre Nodes

It is well known that the Lagrange interpolation of a continuous function based on the Chebyshev nodes may be divergent everywhere (for arbitrary nodes, almost everywhere) like the Fourier series of a summable function. On the other hand any measurable almost everywhere finite function can be “adjusted” in a set of arbitrarily small measure such that its Fourier series will be uniformly convergent. The question arises: does the class of continuous functions have a similar property with respect to any interpolation process?

## Connections of Nonzero Curvature on Three-dimensional Non-reductive Spaces

When a homogeneous space admits an invariant affine connection? If there exists at least one invariant

connection then the space is isotropy-faithful, but the isotropy-faithfulness is not sufficient for the space in

order to have invariantconnections. If a homogeneousspace is reductive, then the space admits an invariant

connection.Thepurposeoftheworkisadescriptionofthree-dimensionalnon-reductivehomogeneousspaces,

admitting invariant affine connections of nonzero curvature only, and the affine connections, curvature and

## Approximation of Control for Singularly Perturbed System with Delay with Integral Quadratic Constraints

The purpose of the work is the development and theoretical substantiation of analytical approximate or asymptotic methods for solving optimal control problems for singularly

perturbed systems with constant delay in phase variables under conditions of uncertainty with respect to the initial data. For achievement of a goal the control problem for the singularly

perturbed system with delay with indeterminate initial conditions and integral quadratic constraints on the control resources according to the minimax criterion is considered. A limit problem

## On Almost Nilpotent Varieties with Integer PI-Exponent

We study almost nilpotent varieties of algebras over a field of zero characteristic. Earlier in the class of algebras with identical relation x(yz) ≡ 0 and in the class of all commutative metabelian algebras countable sets of varieties with integer PI-exponent were defined. Only the existence of almost nilpotent subvariety in each defined variety was proved. In the paper by means of combinatorial methods and methods of the representation theory of symmetric groups we prove that earlier defined varieties are almost nilpotent.

## Equiconvergence Theorem for Integral Operator with Involution

In the paper, the integral operator with kernel having discontinuities of the first kind at the lines t = x and t = 1 − x is studied. The equiconvergence of Fourier expansions for arbitrary integrable function f(x) in eigenfunctions and associated functions of the considered operator and expansions of linear combination of functions f(x) and f(1 − x) in trigonometric system is proved. The equiconvergence is studied using the method based on integration of the resolvent using spectral value. Methods, developed by A. P.

## Generalized Absolute Convergence of Series with Respect to Multiplicative Systems of Functions of Generalized Bounded Variation

A. Zygmund proved that a 2π-periodic function with bounded variation and from any Lipschitz class Lip(α) has absolutely convergent Fourier series. This result was extended to many classes of functions of generalized bounded variation (for example, functions of bounded Jordan-Wiener p-variation, functions of bounded Λ-variation introduced by D. Waterman et al) and to different spaces defined with the help of moduli of continuity.

## On the Geometric Structure of the Continuos Mappings Preserving the Orientation of Simplexes

It is easy to show that if a continuous open map preserves the orientation of allsimplexes, the nit is affine. The class of continuous open maps f : D ⊂ R m → R n that preserve the orientation of simplexes from a given subset of a set of simplexes with vertices in the domain D ⊂ R m is considered. In this paper, questions of the geometric structure of linear inverse images of such mappings are studied.