R.R. Akopyan. Approximation of derivatives of analytic functions from one Hardy class by another Hardy class ... P. 21-29

In the Hardy space $\mathcal{H}^p(D_\varrho)$, $1\le p\le\infty$, of functions analytic in the disk $D_\varrho=\left\{z\in\mathbb{C}\,:\,|z|<\varrho\right\}$, we denote by $NH^p(D_\varrho)$, $N>0$, the class of functions whose $L^p$-norm on the circle $\gamma_\varrho=\left\{z\in\mathbb{C}\, :\, |z|=\varrho\right\}$ 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:
$$ \mathcal{E}\left(\partial H^p(D_\rho), NH^p(D_R)\right)_{L^p(\Gamma_r)} \asymp N^{-\beta/\alpha} \ln^{1/\alpha}N, \quad \alpha=\frac{\ln R-\ln\rho}{\ln R-\ln r}, \quad \beta=1-\alpha. $$
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

Received April 1, 2019

Revised May 7, 2019

Accepted May 13, 2019

Funding Agency: 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. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Roman Razmikovich Akopyan, Cand. Sci. (Phys.-Math.), Ural Federal University, Yekaterinburg, 620002 Russia; Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; e-mail: RRAkopyan@mephi.ru

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Cite this article as: R.R.Akopyan. Approximation of derivatives of analytic functions from one Hardy class by another Hardy class, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 2, pp. 21–29.