Summation Methods Of Fourier Series Research Articles

Approximation of Functions of the Classes CβψHα by Linear Methods Summation of Their Fourier Series

In this paper, we considered arbitrary linear summation methods of Fourier series specified by a set of continuous functions dependent on the real parameter and established their approximation properties. We obtained asymptotic formulas for the exact upper bounds of the deviations of operators generated by λ-methods of Fourier series summation from the functions of the classes CβψHα under certain restrictions on the functions ψ.

Approximation properties of Abel-Poisson integrals on the classes of differentiable functions, defined by means of modulus of continuity

Being the natural apparatus of the periodic functions approximation, the partial Fourier sums are not uniformly convergent over the entire space of the continuous functions. This fact stimulated the search for ways to construct sequences of polynomials that would converge uniformly on the entire space. The matrix method of Fourier series summation is one of the most common methods. Many results on the approximation of the classes of differentiated functions have been obtained for methods generated by triangular infinite matrices. The set of approximating linear methods can be extended by the process of summation of Fourier series, when instead of an infinite triangular matrix one considers the set $\Lambda=\{\lambda_{\delta}(k)\}$ of functions of the natural argument depending on the real parameter $\delta$. The paper deals with the problem of approximation in the uniform metric of $W^{1}H_{\omega}$ classes using one of the classical linear summation methods for Fourier series given by a set of functions of a natural argument, namely, using the Abel-Poisson integral. At the same time, emphasis is placed on the study of the asymptotic behavior of the exact upper limits of the deviations of the Abel-Poisson integrals from the functions of the mentioned class.

Approximation of functions by linear summation methods in the Orlicz-type spaces

Approximative properties of linear summation methods of Fourier series are considered in the Orlicz-type spaces SM. In particular, in terms of approximations by such methods, constructive characteristics are obtained for the classes of functions whose moduli of smoothness do not exceed a certain majorant.

Approximation of functions by linear summation methods in the Orlicz type spaces

Approximative properties of linear summation methods of Fourier series are considered in the Orlicz type spaces S_M. In particular, in terms of approximations by such methods, constructive characteristics are obtained for classes of functions whose moduli of smoothness do not exceed a certain majorant.

Summation Methods for Fourier Series with Respect to the Azoff–Shehada System

A special class of complete minimal systems with complete biorthogonal system in a Hilbert space is considered. This class was introduced by Azoff and Shehada. The paper studies conditions under which there exists a linear summation method for Fourier series with respect to the Azoff–Shehada system. A construction of a linear summation method of the Fourier series for a given vector is presented, as well as a construction of a universal linear summation method.

On Saturation of Linear Summation Methods for Fourier Series in the Spaces Sjp

We consider the problem of the saturation of linear summation methods for Fourier series in the spaces \(S_\varphi ^p \), p > 0. We show that the saturation of a linear method and the saturation order are independent of the parameters X, ϕ, and p that define the space \(S_\varphi ^p \)(X).

Approximation of classes of periodic functions with small smoothness

We prove that the approximations of classes of periodic functions with small smoothness in the metrics of the spaces C and L by different linear summation methods for Fourier series are asymptotically equal to the least upper bounds of the best approximations of these classes by trigonometric polynomials of degree not higher than (n - 1). We establish that the Fejer method is asymptotically the best among all positive linear approximation methods for these classes.

Some linear summation methods for Fourier series

New non-traditional methods are given that ensure uniform convergence for summation of trigonometric Fourier series of continuous functions.

Summation methods for fourier series yielding best order of approximation

Summation methods for fourier series yielding best order of approximation

Multiple fourier series and integrals

One gives a brief survey of the investigations on the theory of multiple Fourier series and integrals, reviewed in Referativnyi Zhurnal “Matematika” in the period 1953–1980. Principal attention is given to the following questions: localization principles, uniform convergence and summability, convergence and summability at a point, in the Lp metric, and almost everywhere, absolute convergence, uniqueness theorems, conjugate Fourier series and integrals, equiconvergence and equisummability of Fourier series and integrals, properties of the kernel and of the Lebesgue constant of summation methods of Fourier series, Fourier coefficients and Fourier transforms.

On linear summation methods of Fourier series

Получены новые оценк иL-нормы тригонометр ических полиномов $$T_n (t) = \frac{{\lambda _0 }}{2} + \mathop \sum \limits_{k = 1}^n \lambda _k \cos kt$$ в терминах коэффицие нтовλ k и их разностейΔλ k=λ k−λ k−1: (1) $$\mathop \smallint \limits_{ - \pi }^\pi |T_n (t)|dt \leqq \frac{c}{n}\mathop \sum \limits_{k = 0}^n |\lambda _\kappa | + c\left\{ {x(n,\varphi )\mathop \sum \limits_{k = 0}^n \Delta \lambda _\kappa \mathop \sum \limits_{l = 0}^n \Delta \lambda _l \delta _{\kappa ,l} (\varphi )} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ,$$ где $$\kappa (n,\varphi ) = \mathop \smallint \limits_{1/n}^\pi [t^2 \varphi (t)]^{ - 1} dt, \delta _{k,1} (\varphi ) = \mathop \smallint \limits_0^\infty \varphi (t)\sin \left( {k + \frac{1}{2}} \right)t \sin \left( {l + \frac{1}{2}} \right)t dt,$$ a ϕ(t) — произвольная фун кция ≧0, для которой опр еделены соответствующие инт егралы. Из (1) следует, что методы $$\tau _n (f;t) = (N + 1)^{ - 1} \mathop \sum \limits_{k = 0}^{\rm N} S_{[2^{k^\varepsilon } ]} (f;t), n = [2^{N\varepsilon } ],$$ являются регулярным и для всех 0<e≦1/2. ЗдесьS m (f, x) частные суммы ряда Фу рье функцииf(x). В статье исследуется многомерный случай. П оказано, что метод суммирования (о бобщенный метод Рисса) с коэффиц иентами $$\lambda _{\kappa ,l} = (R^v - k^\alpha - l^\beta )^\delta R^{ - v\delta } (0 \leqq k^\alpha + l^\beta \leqq R^v ;\alpha \geqq 1,\beta \geqq 1,v< 0)$$ является регулярным, когда δ > 1.

The approximation of periodic functions by linear summation methods for Fourier series

The approximation of periodic functions by linear summation methods for Fourier series