# Mathematics

## Approximation and Reconstruction of Continuous Function with Boundary Conditions

This work deals with a family of integral operators, which are used to get uniform approximations to continuous function with boundary conditions (stated approximations with the same conditions as well); the Kolmogorov – Nikolsky problem is solved on some compact class. Acquired problem from the theory of ill-posed problems (so-called problem of reconstruction of a continuous function using its mean-root-square approximation) is solved via the goal family of integral operators as well.

## Regularization of Abel Equation with the Use of Discontinuous Steklov Operator

For getting uniform approximations of the exact solution of Abel equation with an approximate right-hand part a simply constructed family of integral operators is suggested.

## Approximation of Function and Its Derivative by the Modificated Steklov Operator

With the use of modification of Steklov operator are constructed families of integral operator which allow us to get uniform derivative on a closed.

## About the Norms of Interpolation Processes with Fixed Nodes

The object of study is interpolating rational Lagrange functions. The aim of the research — the study of approximation properties of these functions in the space of square integrated functions. In the introduction the relevance of the research is indicated, references to some works related to this article are given. We also describe the construction of the apparatus of approximation — interpolating rational Lagrange functions. In the main part the norm of the interpolating rational function in the space of the square integrated functions is calculated.

## On Spectrum of Schrödinger Operator on Manifold of a Special Type

The main subject of the paper is spectrum of the Schrödinger operator on weighted quasimodel manifold with an end, which is warped product of a special type. We prove the criterion of discreteness for the spectrum of the operator in terms of metric coefficients and potential of the operator. As the conclusion we made some remarks on the corollaries of the proved theorem and on its extension to more complex quasimodel manifolds.

## On Multiple Completeness of the Root Functions of a Certain Class of Pencils of Differential Operators with Constant Coefficients

A polinomial pencil of ordinary differential operators of *n*-th order generated by a homogeneous differential expression with constant coefficients and by two-point boundary conditions of a special structure with *l *conditions in zero only (1 ≤ *l *≤ *n*−1) is considered in the space *L*2[0*,*1]. The case is studied, when the roots of the characteristic equation lie on a ray coming from the origin.

## Martingales and Theorems of Cantor–Young–Bernstein and de la Vallée Poussin

Uniqueness problems for one-dimensional Haar series and for multiple ones have understood in numerous works. It is well-known that the subsequence of the partial sums S_{2k} of an arbitrary Haar series can be represented as a discrete-time martingale on some filtered probability space (Ω, F, (F_{k}), P). In paper the concept of a U -set for martingales is presented and some uniqueness theorems for martingales on arbitrary compact filtered probability spaces are established.

## Riescz Basis Property of Eigen and Associated Functions of Integral Operators with Discontinuous Kernels, Containing Involution

For invertible integral operator which kernel is discontinuous on the diagonals of the unit square Riescz basis property of its eigen and associated functions in L_{2}[0, 1] is proved.

## Green Function of the Dirichlet Boundary Value Problem for Polyharmonic Equation in a Ball Under Polynomial Data

The classical Dirichlet boundary value problem for the polyharmonic equation in the unit ball is considered. For this problem with polynomial right-hand side and zero boundary data a polynomial solution is constructed. Our approach is based on the Almansi representation of polyharmonic functions and on the previously obtained an explicit representation of the harmonic components, expressed through the given polyharmonic function. In the case of the harmonic equation the known representation of the solution through the Green function is obtained.

## Uniqueness of Solution of the Inverse Scattering Problem for Various Order Differential Equation on the Simplest Noncompact Graph with Cycle

An inverse scattering problem is studied for variable orders differential operators on simplest noncompact graph with cycle. A uniqueness theorem of recovering coefficients of operators from the scattering data is provided.