Izvestiya of Saratov University.
ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


Jordan–Dirichlet Theorem for Functional Differential Operator with Involution

In this paper the problem of decomposability of a function f(x) into Fourier series with respect to the system of eigenfunctions of a functional-differential operator with involution Ly = y′(1 − x) + ®y′(x) + p1(x)y(x) + p2(x)y(1−x), y(0) = °y(1) is investigated. Based on the study of the resolvent of the operator easier and using the method of contour integration of the resolvent, we obtain the sufficient conditions for the convergence of the Fourier series for a function f(x) (analogue of the Jordan–Dirichlet’s theorem).

Integral operator with kernel having jumps on broken lines

 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. 

Operator Integration with an Involution Having a Power Singularity

Spectral properties of the integral operator with an involution of special type in the upper limit are studied and an equiconvergence theorem for its generalized eigenfunction expansions is obtained.

The Theorem on Equiconvergence for the Integral Operator on Simplest Graph with Cycle

The paper deals with integral operators on the simplest geometric two-edge graph containing the cycle. The class of integral operators with range of values satisfying continuity condition into internal node of graph is described. The equiconvergence of expansions in eigen and adjoint functions and trigonometric Fourier series is established.

On the Same Theorem on a Equiconvergence at the Whole Segment for the Functional Differential Operators

The equiconvergence of expansions in eigen- and adjoint functions of functional-differential operator with involution, containing the potentials, and simplest functional-differential operator at the whole segment of Fourier series is established.