Complete asymptotics of the approximation of function from the Sobolev classes by the Poisson integrals

In the paper, we investigate an asymptotic behavior of the sharp upper bounds in the integral metric of deviations of the Abel-Poisson integrals from functions from the class $L^{\psi}_{\beta, 1}$. The Abel-Poisson integrals are solutions of the partial differential equations of elliptic type with corresponding boundary conditions, and they play an important role in applied problems. The approximative properties of the Abel-Poisson integrals on different classes of differentiable functions were studied in a number of papers. Nevertheless, a problem on the respective approximation on the classes $L^{\psi}_{\beta,1}$ in the metric of the space $L$ remained unsolved. We managed to obtain the estimates for the values of approximation of $(\psi, \beta)$-differentiable functions from the unit ball of the space $L$ by the Abel-Poisson integrals. In some cases, we also write down asymptotic equalities for these quantities, that is we solve the Kolmogorov-Nikol'skii problem for the the Abel-Poisson integrals on the classes $L^{\psi}_{\beta,1}$ in the integral metric.

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“…Taking into account the estimate (60) from [13], we derive to (5). Hence, ( 7), (20), ( 4) and ( 5) prove the statement of the theorem.…”

Section: Approximation Of (ψ β)-Differentiable Functions By the Abel-...mentioning

confidence: 56%

On approximation of functions from the class $L^{\psi}_{\beta, 1}$ by the Abel-Poisson integrals in the integral metric

Zhyhallo

Kharkevych

2022

Carpathian Math. Publ.

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“…Note that approximative properties of 2π-periodic analogs of these kernels were considered in the papers [9,10]. Denote by…”

Section: Introductionmentioning

confidence: 99%

The use of the isometry of function spaces with different numbers of variables in the theory of approximation of functions

Bushev

Аbdullayev

Kal’chuk

et al. 2021

Carpathian Math. Publ.

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In the work, we found integral representations for function spaces that are isometric to spaces of entire functions of exponential type, which are necessary for giving examples of equality of approximation characteristics in function spaces isometric to spaces of non-periodic functions. Sufficient conditions are obtained for the nonnegativity of the delta-like kernels used to construct isometric function spaces with various numbers of variables.

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“…Что касается общей теории линейных методов суммирования рядов Фурье [7], то особое место среди последних занимают методы, которые задаются посредством совокупности функций натурального аргумента [8][9][10][11][12][13][14]. Примеры таких линейных методов -методы суммирования Абеля-Пуассона [15][16][17] (или же просто Пуассона [18][19][20]), Гаусса-Вейерштрасса [21], бигармонического [22][23][24] и тригармонического [25][26][27] интегралов Пуассона.…”

Section: Introductionunclassified

О Приближении Функций Класса Зигмунда Бигармоническими Интегралами Пуассона

Borsuk¹,

Khanin²

2021

PC&I

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The paper is devoted to a behavior investigation of the upper bound of deviation of functions from Zygmund classes from their biharmonic Poisson integrals. Systematic research in this direction was conducted by a number of Ukrainian as well as foreign scientists. But most of the known results relate to an estimation of deviations of functions from different classes from operators that were constructed based on triangular l-methods of the Fourier series summation (Fejer, Valle Poussin, Riesz, Rogozinsky, Steklov, Favard, etc.). Concerning the results relating to linear methods of the Fourier series summation, given by a set of functions of natural argument (Abel-Poisson, Gauss-Weierstrass, biharmonic and threeharmonic Poisson integrals), in this direction the progress was less notable. This may be due to the fact that the above-mentioned linear methods the Fourier series summation are solutions of corresponding integral and differential equations of elliptic type. And, therefore, they require more time-consuming calculations in order to obtain some estimates, that are suitable for a direct use for applied purposes. At the same time, in the present paper we investigate approximative characteristics of linear positive Poisson-type operators on Zygmund classes of functions. According to the well-known results by P.P. Korovkin, these positive linear operators realize the best asymptotic approximation of functions from Zygmund classes. Thus, the estimate obtained in this paper for the deviation of functions from Zygmund classes from their biharmonic Poisson integrals (the least studied and most valuable among all linear positive operators) is relevant from the viewpoint of applied mathematics.

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Complete asymptotics of the approximation of function from the Sobolev classes by the Poisson integrals

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Cited by 18 publications

References 9 publications

On approximation of functions from the class $L^{\psi}_{\beta, 1}$ by the Abel-Poisson integrals in the integral metric

On approximation of functions from the class $L^{\psi}_{\beta, 1}$ by the Abel-Poisson integrals in the integral metric

The use of the isometry of function spaces with different numbers of variables in the theory of approximation of functions

О Приближении Функций Класса Зигмунда Бигармоническими Интегралами Пуассона

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