scholarly journals APPROXIMATION OF CLASSES OF POISSON INTEGRALS BY REPEATED FEJER SUMS

2020 ◽  
Vol 8 (2) ◽  
pp. 114-121
Author(s):  
O. Rovenska
Keyword(s):  
Fourier Series ◽  
Real Variable ◽  
Upper Bounds ◽  
Partial Sums ◽  
Asymptotic Formulas ◽  
Periodic Functions ◽  
Arithmetic Means ◽  
Poisson Integrals ◽  
Fourier Sums ◽  
Uniformly Convergent

The paper is devoted to the approximation by arithmetic means of Fourier sums of classes of periodic functions of high smoothness. The simplest example of a linear approximation of continuous periodic functions of a real variable is the approximation by partial sums of the Fourier series. The sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. A significant number of works is devoted to the study of other approximation methods, which are generated by transformations of Fourier sums and allow us to construct trigonometrical polynomials that would be uniformly convergent for each continuous function. Over the past decades, Fejer sums and de la Vallee Poussin sums have been widely studied. One of the most important direction in this field is the study of the asymptotic behavior of upper bounds of deviations of linear means of Fourier sums on different classes of periodic functions. Methods of investigation of integral representations of deviations of trigonometric polynomials generated by linear methods of summation of Fourier series, were originated and developed in the works of S.M. Nikolsky, S.B. Stechkin, N.P. Korneichuk, V.K. Dzadyk and others. The aim of the work systematizes known results related to the approximation of classes of Poisson integrals by arithmetic means of Fourier sums, and presents new facts obtained for particular cases. In the paper is studied the approximative properties of repeated Fejer sums on the classes of periodic analytic functions of real variable. Under certain conditions, we obtained asymptotic formulas for upper bounds of deviations of repeated Fejer sums on classes of Poisson integrals. The obtained formulas provide a solution of the corresponding Kolmogorov-Nikolsky problem without any additional conditions.

scholarly journals Integral presentation of deviations of rectangular linear means of Fourier series on classes of periodic differentiable functions

2018 ◽  
Vol 32 ◽  
pp. 92-103
Author(s):  
Oleg Novikov ◽  
Olga Rovenska
Keyword(s):  
Fourier Series ◽  
Real Variable ◽  
Upper Bounds ◽  
Integral Representations ◽  
Linear Operators ◽  
Trigonometric Polynomials ◽  
Periodic Functions ◽  
Differentiable Functions ◽  
Fourier Sums ◽  
Linear Means

The paper deals with the problems of approximation in a uniform metric of periodic functions of many variables by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. Questions of asymptotic behavior of the upper bounds of deviations of linear operators generated by the use of linear methods of summation of Fourier series on the classes of periodic differentiable functions are studied in many works. Methods of investigation of integral representations of deviations of polynomials on the classes of periodic differentiable functions of real variable originated and received its development through the works of S.M. Nikol'skii, S.B. Stechkin, N.P.Korneichuk, V.K. Dzadik, A.I. Stepanets, etc. Along with the study of approximation by linear methods of classes of functions of one variable, are studied similar problems of approximation by linear methods of classes of functions of many variables. In addition to the approximative properties of rectangular Fourier sums, are studied approximative properties of other approximation methods: the rectangular sums of Valle Poussin, Zigmund, Rogozinsky, Favar. In this paper we consider the classes of \(\overline{\psi}\)-differentiable periodic functions of many variables, allowing separately to take into account the properties of partial and mixed \(\overline{\psi}\)-derivatives, and given by analogy with the classes of \(\overline{\psi}\)-differentiable periodic functions of one variable. Integral representations of rectangular linear means of Fourier series on classes of \(\overline{\psi}\)-differentiable periodic functions of many variables are obtained. The obtained formulas can be useful for further investigation of the approximative properties of various linear rectangular methods on the classes \(\overline{\psi}\)-differentiable periodic functions of many variables in order to obtain a solution to the corresponding Kolmogorov-Nikolsky problems.

scholarly journals Approximation of the classes $W^{r}_{\beta,\infty}$ by three-harmonic Poisson integrals

2019 ◽  
Vol 11 (2) ◽  
pp. 321-334 ◽  
Author(s):  
U.Z. Hrabova ◽  
I.V. Kal'chuk
Keyword(s):  
Fourier Series ◽  
Function Class ◽  
Upper Bounds ◽  
Extremal Problems ◽  
Asymptotic Formulas ◽  
Specific Method ◽  
Triangular Matrices ◽  
Differentiable Functions ◽  
Poisson Integrals ◽  
Functional Classes

In the paper, we solve one extremal problem of the theory of approximation of functional classes by linear methods. Namely, questions are investigated concerning the approximation of classes of differentiable functions by $\lambda$-methods of summation for their Fourier series, that are defined by the set $\Lambda =\{{{\lambda }_{\delta }}(\cdot )\}$ of continuous on $\left[ 0,\infty \right)$ functions depending on a real parameter $\delta$. The Kolmogorov-Nikol'skii problem is considered, that is one of the special problems among the extremal problems of the theory of approximation. That is, the problem of finding of asymptotic equalities for the quantity $$\mathcal{E}{{\left( \mathfrak{N};{{U}_{\delta}} \right)}_{X}}=\underset{f\in \mathfrak{N}}{\mathop{\sup }}\,{{\left\| f\left( \cdot \right)-{{U}_{\delta }}\left( f;\cdot;\Lambda \right) \right\|}_{X}},$$ where $X$ is a normalized space, $\mathfrak{N}\subseteq X$ is a given function class, ${{U}_{\delta }}\left( f;x;\Lambda \right)$ is a specific method of summation of the Fourier series. In particular, in the paper we investigate approximative properties of the three-harmonic Poisson integrals on the Weyl-Nagy classes. The asymptotic formulas are obtained for the upper bounds of deviations of the three-harmonic Poisson integrals from functions from the classes $W^{r}_{\beta,\infty}$. These formulas provide a solution of the corresponding Kolmogorov-Nikol'skii problem. Methods of investigation for such extremal problems of the theory of approximation arised and got their development owing to the papers of A.N. Kolmogorov, S.M. Nikol'skii, S.B. Stechkin, N.P. Korneichuk, V.K. Dzyadyk, A.I. Stepanets and others. But these methods are used for the approximations by linear methods defined by triangular matrices. In this paper we modified the mentioned above methods in order to use them while dealing with the summation methods defined by a set of functions of a natural argument.

scholarly journals Approximation of classes of Poisson integrals by Fejer sums

2018 ◽  
Keyword(s):  
Complex Plane ◽  
Upper Bounds ◽  
Periodic Functions ◽  
Poisson Integrals

For upper bounds of the deviations of Fejer sums taken over classes of periodic functions that admit analytic extensions to a fixed strip of the complex plane, we obtain asymptotic equalities. In certain cases, these equalities give a solution of the corresponding Kolmogorov-Nikolsky problem.

scholarly journals Approximative characteristics of the Nikol'skii-Besov-type classes of periodic functions in the space $B_{\infty,1}$

2020 ◽  
Vol 12 (2) ◽  
pp. 376-391
Author(s):  
O.V. Fedunyk-Yaremchuk ◽  
M.V. Hembars'kyi ◽  
S.B. Hembars'ka
Keyword(s):  
Linear Operators ◽  
Partial Sums ◽  
Exact Order ◽  
Periodic Functions ◽  
Multivariate Case ◽  
The Best Approximation ◽  
Univariate Case ◽  
Type Classes ◽  
Functional Classes ◽  
Fourier Sums

We obtained the exact order estimates of the orthowidths and similar to them approximative characteristics of the Nikol'skii-Besov-type classes $B^{\Omega}_{p,\theta}$ of periodic functions of one and several variables in the space $B_{\infty,1}$. We observe, that in the multivariate case $(d\geq2)$ the orders of orthowidths of the considered functional classes are realized by their approximations by step hyperbolic Fourier sums that contain the necessary number of harmonics. In the univariate case, an optimal in the sense of order estimates for orthowidths of the corresponding functional classes there are the ordinary partial sums of their Fourier series. Besides, we note that in the univariate case the estimates of the considered approximative characteristics do not depend on the parameter $\theta$. In addition, it is established that the norms of linear operators that realize the order of the best approximation of the classes $B^{\Omega}_{p,\theta}$ in the space $B_{\infty,1}$ in the multivariate case are unbounded.

scholarly journals Asymptotically best possible Lebesgue-type inequalities for the fourier sums on sets of generalized poisson integrals

Filomat ◽  
2020 ◽  
Vol 34 (14) ◽  
pp. 4697-4707
Author(s):  
Anatoly Serdyuk ◽  
Tetiana Stepanyuk
Keyword(s):  
Trigonometric Polynomials ◽  
Periodic Functions ◽  
Best Approximations ◽  
Generalized Poisson ◽  
Poisson Integrals ◽  
Fourier Sums

In this paper we establish Lebesgue-type inequalities for 2?-periodic functions f, which are defined by generalized Poisson integrals of the functions ? from Lp, 1 ? p < 1. In these inequalities uniform norms of deviations of Fourier sums ||f-Sn-1||C are expressed via best approximations En(?)Lp of functions ? by trigonometric polynomials in the metric of space Lp. We show that obtained estimates are asymptotically best possible.

scholarly journals Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Elena E. Berdysheva ◽  
Nira Dyn ◽  
Elza Farkhi ◽  
Alona Mokhov
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Error Bounds ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Dirichlet Kernel ◽  
Partial Sums ◽  
Functions Of Bounded Variation ◽  
Moduli Of Continuity ◽  
Fourier Approximation

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

Sign in / Sign up

Export Citation Format

Share Document