# Mathematics

## The Il’in Spectral Method for Determination of the Properties of the Basis Property and the Uniform Convergence of Biorthogonal Expansions on a Finite Interval

The paper discusses the basics of the spectral method of V. A. Il’in on an example of a simple second order differential operator on a segment of the number line. The ﬁrst theorem of Il’in on the unconditional basis property is stated. Its detailed proof is given. A chain of generalizations of this theorem is traced. A recently established a theorem on the unconditional basis property for the differential operators with general integral boundary conditions is formulated.

## Necessary and Sufficient Condition for an Orthogonal Scaling Function on Vilenkin Groups

There are several approaches to the problem of construction of an orthogonal MRA on Vilenkin groups, but all of them are reduced to the search of the so-called scaling function. In 2005 Yu. Farkov used the so-called “blocked sets” in order to ﬁnd all possible band-limited scaling functions with compact support for each set of certain parameters and his conditions are necessary and sufﬁcient. S. F. Lukomskii, Iu. S. Kruss and G. S.

## Nonlocal Boundary-Value Problems in the Cylindrical Domain for the Multidimensional Laplace Equation

Correct statements of boundary value problems on the plane for elliptic equations by the method of analytic function theory of a complex variable. Investigating similar questions, when the number of independent variablesis greater than two, problems of a fundamental nature arise. Avery attractive and convenient method of singular integral equations loses its validity due to the absence of any complete theory of multidimensional singular integral equations. The author has previously studied local boundary value problems in a cylindrical domain for multidimensional elliptic equations.

## Approximation of Continuous 2 p-Periodic Piecewise Smooth Functions by Discrete Fourier Sums

Let N be a natural number greater than 1. Select N uniformly distributed points tk = 2πk/N + u (0 6 k 6 N − 1), and denote by Ln,N(f) = Ln,N(f,x) (1 6 n 6 N/2) the trigonometric polynomial of order n possessing the least quadratic deviation from f with respect to the system {tk}N−1 k=0 . Select m + 1 points −π = a0 < a1 < ... < am−1 < am = π, where m > 2, and denote Ω = {ai}m i=0. Denote by Cr Ω a class of 2π-periodic continuous functions f, where f is r-times differentiable on each segment ∆i = [ai,ai+1] and f(r) is absolutely continuous on ∆i.

## On Inverse Problem for Differential Operators with Deviating Argument

Second-order functional differential operators with a constant delay are considered. Properties of their spectral characteristics are obtained, and a nonlinear inverse spectral problem is studied, which consists in constructin goperators from the irspectra. We establish the unique nessand develop a constructive procedure for solution of the inverse problem.

## Hermite Interpolation on a Simplex

In the paper, we solve the problem of polynomial interpolation and approximation functions of several variable sonann dimensional simplex in the uniform normus ingpoly nomials of the third degree.Wechoose interpolation conditions in terms of derivatives in the directions of the edges of a simplex. In the same terms we obtained estimates of the deviation of derivatives of polynomial from the corresponding derivatives of an interpolated function under the assumption that the interpolated function has continuous directional derivatives up to the fourth order inclusive.

## Some Properties of 0/1-Simplices

Let n ∈ N, and let Q n = [0,1] n . For a nondegenerate simplex S ⊂ R n , by σS we mean the homothetic copy of S with center of homothety in the center of gravity of S and ratio of homothety σ. Put ξ(S) = min{σ > 1 : Q n ⊂ σS}, ξ n = min{ξ(S) : S ⊂ Q n }.

## Criterion for a Generalized Solution in the Class Lp for the Wave Equation to Be in the Class W1 p

In this paper we consider the question of whether a generalized solution of the wave equation belongs to different function spaces. Consideration of classical solutions imposes substantial restrictions on the initial data of the problem. But if we proceed not from differential but from integral equations, then the class of solutions and the class of initial boundary value problems can be substantially expanded. To solve the boundary value problem for the wave equation obtained by the wave counting method, it is easy to obtain a sufficient condition for belonging to a particular class.

## Non-reductive Homogeneous Spaces Not Admitting Normal Connections

The purpose of the work is the classification of three-dimensional non-reductive homogeneous spacesnot admitting normal connections, affine connections, their torsion tensors, curvature and holonomy algebras.The object of investigation arepointed-non-reductive spaces and connections on them. The basic notions, such as the isotropically-faithful pair, reductive space, afne connection, curvature tensor and torsion tensor, holonomy algebra and normal connection are defined.

## An Asymptotic Relation for Conformal Radii of Two Nonoverlapping Domains

We consider a family of continuously varying closed Jordan curves given by a polar equation, such that the interiors of the curves form an increasing or decreasing chain of domains. Such chains can be described by the Löwner–Kufarev differential equation. We deduce an integral representation of a driving function in the equation.Using this representation we obtainan a symptotic formula, which establishes a connection between conformal radii of bounded and unbounded components of the complement of the Jordan curve when the bounded component is close to the unit disk.