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Approximation of derivatives of analytic functions from one Hardy class by another Hardy class
R. R. Akopyan^{ab} ^{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.
DOI:
https://doi.org/10.21538/0134488920192522129
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UDC:
517.977
MSC: 30E10, 30H10 Received: 01.04.2019
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 2129
\mathnet{http://mi.mathnet.ru/timm1620}
\crossref{https://doi.org/10.21538/0134488920192522129}
\elib{https://elibrary.ru/item.asp?id=38071595}
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This publication is cited in the following articles:

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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