собственные функции

The Principle of Localization at the Class of Functions Integrable in the Riemann for the Processes of Lagrange –Sturm – Liouville

Authors: Trynin A. Y., Kireeva E. D.

Let us say that the principle of localization holds at the class of functions F at point x0 ∈ [0, π] for the Lagrange –Sturm – Liouville interpolation process L SL n (f, x) if limn→∞ L SL n (f, x0) − L SL n (g, x0) = 0 follows from the fact that the condition f(x) = g(x) is met for any two functions f and g belonging to F in some neighborhood Oδ(x0), δ > 0.

Equiconvergence Theorem for Integral Operator with Involution

Authors: Nazarova E. V., Khalova V. A.

In the paper, the integral operator with kernel having discontinuities of the first kind at the lines t = x and t = 1 − x is studied. The equiconvergence of Fourier expansions for arbitrary integrable function f(x) in eigenfunctions and associated functions of the considered operator and expansions of linear combination of functions f(x) and f(1 − x) in trigonometric system is proved. The equiconvergence is studied using the method based on integration of the resolvent using spectral value. Methods, developed by A. P.

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.

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.

On Propertiesofthe Eigenfunctionsofa Quadratic Pencilofthe Second Order Differential Operators

Authors: Rykhlov V. S.

The degenerated second order ordinary differential quadratic pencil with constant coefficients is considered. The case is studied, when the roots of characteristic equation lie on a straight line coming through the origin and on the both side of the origin. Properties of the system of its eigenfunctions in the spaces L2[0,σ], σ > 0 is investigated. Criteria of one-fold completeness and minimality of this systemareprovedandsufficientconditionsofone-foldcompleteness and minimality are found.

О сходимости разложений по собственным функциям интегральных операторов с разрывным ядром

Authors: Kornev V. V.

Для интегральных операторов со скачком ядра на диагонали найдены необходимые и достаточные условия их обратимости. Установлено условие,обеспечивающее равносходимость рядов Фурье по собственным функциям этих операторов и тригонометрических рядов Фурье. 

Non-stationary vibration of growth circular cylindrical shell

Authors: Baryshev A. A., Manzhirov A. V., Lychev S. A.

Small forced vibrations of growing cylindrical shell fixed on circular boundaries is studied in the framework of Kirchhoff–Love shell theory. The process of the accretion are characterized by the continuous adherence of material particles to its facial surface. Since the shell bends during the accretion, its stressed-strained state depends not only on loading, but also on the history of the process of accretion, i.e. the schedule of accretion.