ряды Фурье

Approximation Properties of Dicrete Fourier Sums for Some Piecewise Linear Functions

Authors: Akniev G. G.

Let N be a natural number greater than 1. We select N uniformly distributed points t_k = 2πk/N (0 < k < N − 1) on [0,2\pi]. Denote by  L_ n,N (f) = L _n,N (f,x)1 < n < ⌊N/2⌋  the trigonometric polynomial of order n possessing the least quadratic deviation from f with respect to the system tk{k=0}^{N-1}. In other words, the greatest lower bound of the sums on the set of trigonometric polynomials Tn of order n is attained by L_n,N (f). In the present article the problem of function approximation by the polynomials L_n,N (f,x)  is considered.

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. 

Limit Discrete Meixner Series and Their Approximative Properties

Authors: Sultanov E. S.

In this article the problemof function approximation by discrete series by Meixner polynomials orthogonal on uniform net {0, 1, . . .} is investigated. We constructed new series by these polynomials for which partial sums coincidewith input function f(x) in x = 0. These new series were constructed by the passage to the limit of Fourier series Σk=0∞fαkmαk(x) by Meixner polynomials when α → −1.

Discrete Transform with Stick Property Based on {sinx sinkx} and Second Kind Chebyshev Polynomials Systems

Authors: Sharapudinov I. I., Akniev G. G.

In this paper we introduce the discrete series with the «sticking»-property of the periodic ({sinx sinkx} system) and non-periodic (using the system of the second kind of Chebyshev polynomials Uk(x)) cases. It is shown that series of the system {sinx sinkx}
have an advantage over cosine Fourier series because they have better approximation properties near the bounds of the [0, π] segment. Similarly discrete series of the system Uk(x) near the bound of the [−1, 1] approximates given function significantly

The Intermediate Case of Regularity in the Problem of Differentiation of Multiple Integrals

Authors: Fufaev D. V.

The paper deals with generalization of Lebesgue and Jessen –Marcinkiewicz – Zygmund theorems of the differentiation of multiple integrals for the intermediate case of regularity of the system of sets. The application of the result to the Fourier-Haar series and to orthorecursive expansions with respect to system of indicators of multi-dimensional intervals is considered.

The One-dimensional Problem of Unsteady-related Elastic Diffusion Layer

Authors: Gachkevich A. R., Zemskov A. V., Tarlakovsky D. V.

The problem of determining the stress strain state of an elastic medium, taking into account the structural changes caused by the presence of diffusion fluxes. The influence of diffusion processes on the stress-strain state of the environment is taken into account by using the locally equilibrium model of thermoelastic diffusion, which includes the coupled system of equations of motion of an