# Mathematics

## An Inverse Problem for Quasilinear Elliptic Equations

The article examines incorrect return problems in the defining unknown factors in the quasilinear elliptic equation. Theorems of existence, uniqueness and stability have been proved. The consecutive approach method is used for the construction of the regulating algorithm for defining several factors.

## Recovering Differential Operators on a Graph with a Cycle and with Generalized Matching Conditions

The solution of the inverse spectral problem is obtained for second-order differential operators on a graph with a cycle and with generalized matching conditions in the internal vertex.

## On Convergence of Fourier – Vilenkin Series in L p [0, 1), 0 < p ≤ 1

In this paper we study convergence a.e. and L p -convergence (0 < p ≤ 1) of Fourier –Vilenkin series under some tauberian conditions on Fourier coefficients of a function. In the case of Fourier – Walsh series these results are obtained by F. Moricz.

## Approximative Properties of Mixed Series by Lagerre’s Polynomials on Classes of Smooth Functions

Approximative properties of mixed series by Lagerre’s polynomials on classes of smooth functions that given on axle [0, ∞) are viewed. Inequality that corresponds to Lebesgue inequality for trigonometric Fourier sums was found for evaluation of deflection of smooth function from it’s partial sums of mixed series by Lagerre’s polynomials. Evaluations for corresponding Lebesgue function of partial sums of mixed series by Lagerre’s polynomials were found.

## On Inverse Nodal and Spectral Problems for Boundary Value Problems with Discontinuity Conditions Inside the Interval

The solution of inverse nodal and inverse spectral problems is presented for second-order differential operators on a finite interval with discontinuity conditions inside the interval. Connections between these two classes of inverse problems are established.

## About Approximation Multinominals, Orthogonal on Any Grids

In this work are investigated approximation properties of multinominals pˆn(x), orthogonal with weight ∆tj on the any grids consisting of final number of points of a piece [−1, 1]. Namely the approximation formula, in which is established at increase n together with N, approximation behaviour of these multinominals close to approximation behaviour of multinominals Lasiandra.

## O РАЗРЕШИМОСТИ ОБРАТНОЙ ЗАДАЧИ ШТУРМА – ЛИУВИЛЛЯ В СИММЕТРИЧНОМ СЛУЧАЕ

Necessary and sufficient conditions are provided for the solvability of the inverse problem of recovering Sturm – Liouville operator from its spectrum in the central symmetry case.

## On Laguerre Expansions Summability by the Linear Methods

This paper studies a problem of Laguerre expansions summability via methods defined by triangular matrices. The conditions on a matrix and an expandable function are obtained to guarantee the convergence of corresponding linear means at the Lebesgue point t = 0.

## On the Equiconvergence of Expansions for the Certain Class of the Functional-Differential Operators with Involution on the Graph

The equiconvergence of expansions in eigen- and adjoint functions and trigonometric Fourier series is established for a 1-st order functional-differential operator on the graph-cycle.

## Three-Element Boundary Value Problem of Riemann Type for Metaanalytical Functions in a Circle

The article is devoted to the investigation of three-element boundary value problem of Riemann type for metaanalytical functions. A constructive method for solution of the problem in a circle was found. It is established that solution of the problem generally consists of solutions of two generalized and two usual scalar boundary value problems of Riemann for analytical functions in a circle.