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 Trudy Inst. Mat. i Mekh. UrO RAN, 2019, Volume 25, Number 2, Pages 21–29 (Mi timm1620)

Approximation of derivatives of analytic functions from one Hardy class by another Hardy class

R. R. Akopyanab

a Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
b Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg

Abstract: In the Hardy space $\mathcal{H}^p(D_\varrho)$, $1\le p\le\infty$, of functions analytic in the disk $D_\varrho=ż\in\mathbb{C} : |z|<\varrho\}$, we denote by $NH^p(D_\varrho)$, $N>0$, the class of functions whose $L^p$-norm on the circle $\gamma_\varrho=ż\in\mathbb{C} : |z|=\varrho\}$ does not exceed the number $N$ and by $\partial H^p(D_\varrho)$ the class consisting of the derivatives of functions from $1H^p(D_\varrho)$. We consider the problem of the best approximation of the class $\partial H^p(D_\rho)$ by the class $NH^p(D_R)$, $N>0$, with respect to the $L^p$-norm on the circle $\gamma_r$, $0<r<\rho<R$. The order of the best approximation as $N\rightarrow+\infty$ is found:
$$\mathcal{E}(\partial H^p(D_\rho), NH^p(D_R))_{L^p(\Gamma_r)} \asymp N^{-\beta/\alpha} \ln^{1/\alpha}N, \quad \alpha=\frac{\ln R-\ln\rho}{\ln R-\ln r}, \quad \beta=1-\alpha.$$
In the case where the parameter $N$ belongs to some sequence of intervals, the exact value of the best approximation and a linear method implementing it are obtained. A similar problem is considered for classes of functions analytic in rings.

Keywords: analytic functions, Hardy class, best approximation of a class by a class.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-01-00336 Ministry of Education and Science of the Russian Federation 02.A03.21.0006 This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 18-01-00336 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

DOI: https://doi.org/10.21538/0134-4889-2019-25-2-21-29

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Bibliographic databases:

UDC: 517.977
MSC: 30E10, 30H10

Citation: R. R. Akopyan, “Approximation of derivatives of analytic functions from one Hardy class by another Hardy class”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, no. 2, 2019, 21–29

Citation in format AMSBIB
\Bibitem{Ako19} \by R.~R.~Akopyan \paper Approximation of derivatives of analytic functions from one Hardy class by another Hardy class \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2019 \vol 25 \issue 2 \pages 21--29 \mathnet{http://mi.mathnet.ru/timm1620} \crossref{https://doi.org/10.21538/0134-4889-2019-25-2-21-29} \elib{https://elibrary.ru/item.asp?id=38071595} 

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This publication is cited in the following articles:
1. R. R. Akopyan, “Optimal recovery of a derivative of an analytic function from values of the function given with an error on a part of the boundary. II”, Anal. Math., 46:3 (2020), 409–424
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