Fourier series

Approximation of Continuous 2 p-Periodic Piecewise Smooth Functions by Discrete Fourier Sums

Authors: Akniev G. G.

Let N be a natural number greater than 1. Select N uniformly distributed points tk = 2πk/N + u (0 6 k 6 N − 1), and denote by Ln,N(f) = Ln,N(f,x) (1 6 n 6 N/2) the trigonometric polynomial of order n possessing the least quadratic deviation from f with respect to the system {tk}N−1 k=0 . Select m + 1 points −π = a0 < a1 < ... < am−1 < am = π, where m > 2, and denote Ω = {ai}m i=0. Denote by Cr Ω a class of 2π-periodic continuous functions f, where f is r-times differentiable on each segment ∆i = [ai,ai+1] and f(r) is absolutely continuous on ∆i.

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.

Harmonic Analysis of Periodic at Infinity Functions from Stepanov Spaces

Authors: Strukova I. I.

We consider Stepanov spaces of functions defined on R with their values in a complex Banach space. We introduce the notions of slowly varying and periodic at infinity functions from Stepanov space. The main results of the article are concerned with harmonic analysis of periodic at infinity functions from Stepanov space. For this class of functions we introduce the notion of a generalized Fourier series; the Fourier coefficients in this case may not be constants, they are functions that are slowly varying at infinity.

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.

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.

About harmonic analysis of periodic at infinity functions

Authors: Strukova I. I.

We consider slowly varying and periodic at infinity multivariable functions in Banach space. We introduce the notion of Fourier series of periodic at infinity function, study the properties of Fourier series and their convergence. Basic results are derived with the use of isometric representations theory. 

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