Approximation of classes of Poisson integrals by Fejer sums

Novikov, O. A.; Rovenska, O. G.

doi:10.20537/2076-7633-2015-7-4-813-819

Cited by 8 publications

(4 citation statements)

References 3 publications

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“…where K(q) is the total elliptic integral of the first kind. Some related problems were studied in [7,8,9,10,11,12]. For q ∈ 0; 2 − √ 3 upper bounds of the deviations of Fejer sums on the classes C q 0,∞ were investigated in [13]…”

Section: Introductionmentioning

confidence: 99%

Approximation of classes of Poisson integrals by Fejer sums

Novikov

Rovenska

Kozachenko

2018

MAMM

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Section: Introductionmentioning

confidence: 99%

Approximation of classes of Poisson integrals by Fejer sums

Novikov

Rovenska

Kozachenko

2018

MAMM

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“…are called Fejer means of function f . Asymptotic equalities for upper bounds of deviations of Fejer means on classes G(q, 0) were obtained in [3,4]:…”

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Approximation of classes of Poisson integrals by Fejer means

Rovenska

2023

MAMM

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The work is devoted to the investigation of problem of approximation of continuous periodic functions by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. The simplest example of a linear approximation of periodic functions is the approximation of functions by partial sums of their Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. Therefore, a many studies is devoted to the research of the approximative properties of approximation methods, which are generated by transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for the whole class of continuous functions. Particularly, Fejer means have been widely studied in the last time. One of the important problems in this field is the study of asymptotic behavior of the upper bounds over a fixed classes of functions of deviations of the trigonometric polynomials. The aim of the work systematizes known results related to the approximation of classes of Poisson integrals of continuous functions by arithmetic means of Fourier sums, and presents new facts obtained for particular cases. The asymptotic behavior of the upper bounds on classes of Poisson integrals of periodic functions of the real variable of deviations of linear means of Fourier series, which are defined by applying the Fejer summation method is studied. The mentioned classes consist of analytic functions of a real variable, which are narrowing of bounded harmonic in unit disc functions of complex variable. In the work, asymptotic equalities for the upper bounds of deviations of Fejer means on classes of Poisson integrals were obtained.

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“…. Ïðîäîâaeåííÿì âèâ÷åííÿ àïðîêñèìàòèâíèõ âëàñòèâîñòåé ëiíiéíèõ ñåðåäíiõ ñóì Ôóð'¹ íà êëàñàõ iíòåãðàëiâ Ïóàññîíà ¹ ðîáîòè [8,9] i [10,11,12], â ÿêèõ îòðèìàíî àñèìïòîòè÷íi ôîðìóëè äëÿ òî÷íèõ âåðõíiõ ìåae âiäõèëåíü íà êëà-…”

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Approximation of Classes of Poisson Integrals by Repeated Fejer Sums

Rovenska¹

2020

BMJ

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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.

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Approximation of classes of Poisson integrals by Fejer sums

Cited by 8 publications

References 3 publications

Approximation of classes of Poisson integrals by Fejer sums

Approximation of classes of Poisson integrals by Fejer sums

Approximation of classes of Poisson integrals by Fejer means

Approximation of Classes of Poisson Integrals by Repeated Fejer Sums

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