Approximation of Derivatives of Analytic Functions from One Hardy Class by Another Hardy Class

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 annuli.