собственные и присоединенные функции

Multiple Completeness of the Root Functions of the Pencils of Differential Operators with Constant Coefficients and Splitting Boundary Conditions

Authors: Rykhlov V. S.

In the space of square summable functions on the main segment [0,1], the class of polynomial pencils of ordinary differential operators of the n-th order is considered. The coefficients of the differential expression are assumed to be constants. The boundary conditions are assumed to be splitting and two-point at the ends 0 and 1 (l of boundary conditions is taken only at the point 0, and the remaining n − l is taken at the point 1). The differential expression and the boundary forms are assumed to be homogeneous, that is, they contain only main parts.

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

Authors: Lomov I. S.

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.

Multiple Non-Completeness for the System of Eigenfunctions of a Class of the Pencils of Ordinary Differential Operators

Authors: Shigaeva O.

A class of the pencils of ordinary differential operators of n-th order with constant coefficients is considered. The roots of the characteristic equation of the pencils from this class is supposed to lie on a straight line coming through the origin. The main condition is such that the generating functions for the system of eigen- and associatedfunctionsarelinearcombinationsofexponentialfunctions.

О сходимости средних Рисса разложений по собственным и присоединенным функциям интегрального оператора с ядром, имеющим скачки на ломанных линиях

Authors: Koroleva O. A.

В настоящей работе найдены необходимые и достаточные условия равномерной сходимости обобщенных средних Рисса разложений по собственным и присоединенным функциям(с.п.ф.) интегрального оператора, ядро которого терпит скачки на сторонах квадрата, вписанного в единичный квадрат. 

Integral operator with kernel having jumps on broken lines

Authors: Khromov A. P., Koroleva O. A.

 In this paper we study equiconvergence expansions in trigonometric Fourier series, and in eigenfunctions and associated functions of an integral operator whose kernel suffers jumps at the sides of the square inscribed in the unit square.