resolvent

On Riescz Bases of Eigenfunction of 2-nd Order Differential Operator with Involution and Integral Boundary Conditions

Authors: Kurdyumov V. P.

Riesz basisness with brackets of the eigen and associated function is proved for a 2-nd order differential operator with involution in the derivatives and with integral boundary conditions. To demonstrate this the spectral problem of the initial operator is reduced to the spectral problem of a 1-st order operator without involution in the 4-dimensional vector-function space.

About the Classical Solution of the Mixed Problem for the Wave Equation

Authors: Khromov A. P.

The classic solution of the mixed problem for a wave equation with a complex potential and minimal smoothness of initial data is established by the Fourier method. The resolvent approach consists of constructing formal solution with the help of the Cauchy – Poincaré method of integrating the resolvent of the corresponding spectral problem over spectral parameter. The method requires no information about eigen and associated functions and uses only the main part of eigenvalues asymptotics. Krylov’s idea of accelerating the convergence of Fourier series is essentially employed.

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

Authors: Kurdyumov V. P., Khromov A. P.

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 L2[0, 1] is proved.

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. 

An Analogue of the Jordan–Dirichlet Theorem for the Integral Operator with Kernel Having Jumps on Broken Lines

Authors: Koroleva O. A.

In this paper the sufficient conditions (conditions such as Jordan–Dirichlet) expansion function f(x) in a uniformly convergent series of eigenfunctions and associated functions of the integral operator whose kernel is suffering jumps on the sides of the square, inscribed in the unit square. As is known, this expansion requires to f(x) is continuous and belong to the closure of the integral values operator. It turns out that if f(x) also is a function of bounded variation, these conditions are also sufficient.

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