eigenvalues

Mode-Series Expansion of Solutions of Elasticity Problems for a Strip

Authors: Kossovich L. Y., Yurko V. A., Kirillova I. V.

Oscillations of a strip are considered as a plane problem of elasticity theory. Description of oscillation modes is provided. Properties of eigenvalues and eigenfunctions are studied for a boundary value problem for their amplitudes. Green’s function is constructed as a kernel of the inverse operator. Completeness and expansion theorems are proved which allow one to solve problems for finite and infinite membranes under arbitrary boundary conditions.

The Approached Calculation of Eigenvalues of the Discrete Operator by Means of Spectral Traces of Resolvent Degree

Authors: Maleko Е. М.

Letadiscreteself-adjointoperatorT actsinaseparableHilbertspace and have the kernel resolvent, and eigenvalues and eigenfunctions of the operator T be known. In the paper the method of calculation of eigenvalues of the perturbed operator T + P is considered. Resolvent of this operator is presented as convergent Neumann series on eigenfunctions of the operator T. The point of the method is that at first is found a set of numbers which approximate traces of the resolvent degrees of the operator T + P.

On Inverse Problem for Sturm – Liouville Operator with Discontinuous Coefficients

Authors: Akhmedova E. N., Guseinov I. M.

In the paper uniqueness of reconstruction of the Sturm – Liouville operator with discontinuous coefficients by spectral data is proved and algorithm of construction of the potential is provided.

On Riesz Basises of Eigenfunctions of Integral Operators with Kernels Discontinuous on Broken Lines

Authors: Kurdyumov V. P.

For the integral operator, which kernel has jump discontinuities on the sides and diagonals of the four equal subsquares of the unit square 0 ≤ x, t ≤ 1, Riesz basisness of its eigen and associated functions is proved.

Estimates of Speed of Convergence and Equiconvergence of Spectral Decomposition of Ordinary Differential Operators

Authors: Lomov I. S.

The present review contains results of V. A. Il’in and his pupils concerning an assessment of speed of convergence and equiconvergence with a trigonometrical series of Fourier of spectral decomposition of functions on root functions of linear ordinary differential
operators both self-conjugate, and not self-conjugate, set on a final piece of a numerical straight line. The first theorem of V. A. Ilyin of equiconvergence of spectral decomposition for the differential operator of any order is provided. Theorems of the speed of

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.