# Mathematics

## On the Existence of Continual Closed U-set

In this work we consider a system of characters of the Vilrnkin group G and study uniqueness sets for series for system of character of Vilenkin group (in other words, U-sets). We prove a sufficient condition for the U-set on the Vilenkin group and constructed continual closed U-set on the Vilenkin group.

## On Inverse Periodic Problem for Differential Operators for Central Symmetric Potentials

An inverse spectral problem for Sturm–Liouville operators on a finite interval with periodic boundary conditions is studied in the central symmetric case, when the potential is symmetric with respect to the middle of the interval. We discuss the statement of the problem, provide an algorithm for its solution along with necessary and sufficient conditions for the solvability of this nonlinear inverse problem.

## Factorization of Entire Symmetrical Functions of Exponential Type

Let π be an entire function of minimal type of order 1. The entire function F is called π-symmetric if it is represented in the form of a composition f ◦ π, where the f is an entire function. The article deals with the following question. Can we present every π-symmetric function of exponential type as a product of two functions with a close growth, each of which is itself an entire π-symmetric function?

## Special Wavelets Based on Chebyshev Polynomials of the Second Kind and their Approximative Properties

The system of wavelets and scalar functions based on Chebyshev polynomials of the second kind and their zeros is considered. With the help of them we construct a complete orthonormal system of functions.

## About New Approach to Solution of Riemann’s Boundary Value Problem with Condition on the Half-line in Case of Infinite Index

To solve a homogeneous Riemann boundary value problem with infinite index and condition on the half-line we propose a new approach based on the reduction of the considered problem to the corresponding task with the condition on the real axis and finite index.

## Justification of Fourier Method in a Mixed Problem for Wave Equation with Non-zero Velocity

In the paper, using contour integration of the resolvent of the corresponding spectral problem operator, justification of Fourier method in two mixed problems for wave equation with trivial initial function and non-zero velocity is given. The boundary conditions of these problems, together with fixed endpoint conditions, embrace all cases of mixed problems with the same initial conditions for which the corresponding spectral operators in Fourier method have regular boundary conditions. The problems are considered under minimal requirements on initial data. A. N.

## Stochastic Simulation of Diffusion Filtering

Formulated and investigated is the system of kinetic equations describing the process of diffusion filtering based on a stochastic approach. The theorem of existence and uniqueness of the solution for the case of a continuous density is prove.