# R.R. Akopyan, M.S. Saidusainov. Three extremal problems in the Hardy and Bergman spaces of functions analytic in a disk ... P. 22-32.

Let a nonnegative measurable function $\gamma(\rho)$ be nonzero almost everywhere on $(0,1)$, and let the product $\rho\gamma(\rho)$ be summable on$(0,1)$. Denote by $\mathcal{B}=B^{p,q}_{\gamma}$, $1\leq p\le \infty$, $1\leq q < \infty$, the space of functions $f$ analytic in the unit disk for which the function $M_p^q(f,\rho)\rho\gamma(\rho)$ is summable on $(0,1)$, where $M_p^q(f,\rho)$ is the $p$-mean of$f$ on the circle of radius $\rho$; this space is equipped with the norm $\|f\|_{B^{p,q}_{\gamma}}=\|M_p(f,\cdot)\|_{L^q_{\rho\gamma(\rho)}(0,1)}.$ In the case $q=\infty$, the space $\mathcal{B}=B^{p,\infty}_{\gamma}$ is identified with the Hardy space$H^p$. Using an operator$L$ given by the equality $Lf(z)=\sum_{k=0}^\infty l_k c_k z^k$ on functions $f(z)=\sum_{k=0}^\infty c_k z^k$ analytic in the unit disk, we define the class $$LB_\gamma^{p,q}(N):=\{f\colon \|Lf\|_{B_\gamma^{p,q}}\le N\},\quad N>0.$$ For a pair of such operators$L$ and$G$, under some constraints, the following three extremal problems are solved.

(1)The best approximation of the class $LB_\gamma^{p_1,q_1}(1)$ by the class $GB_\gamma^{p_3,q_3}(N)$ in the norm of the space $B_\gamma^{p_2,q_2}$ is found for $2\le p_{1}\le\infty$, $1\leq p_{2}\leq 2$, $1\leq p_{3}\leq 2$, $1\le q_1=q_2=q_3\le\infty$, and $q_s=2$ or $\infty$.

(2)The best approximation of the operator$L$ by the set $\mathcal{L}(N)$, $N>0$, of linear bounded operators from $B_\gamma^{p_1,q_1}$ to $B_\gamma^{p_2,q_2}$ with the norm not exceeding$N$ on the class $GB_\gamma^{p_3,q_3}(1)$ is found for $2\le p_{1}\le\infty$, $1\leq p_{2}\leq 2$, $2\leq p_{3}\leq \infty$, $1\le q_1=q_2=q_3\le\infty$, and $q_s=2$ or $\infty$.

(3)Bounds for the modulus of continuity of the operator$L$ on the class $GB_\gamma^{p_3,q_3}(1)$ are obtained, and the exact value of the modulus is found in the Hilbert case.

Keywords: Hardy and Bergman spaces, best approximation of a class by a class, best approximation of an unbounded operator by bounded operators, modulus of continuity of an operator.

The paper was received by the Editorial Office on May 15, 2017.

Roman Razmikovich Akopyan, Cand. Sci. (Phys.-Math.), Ural Federal University, Ekaterinburg, 620002, Russia; Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia, e-mail: RRAkopyan.mephi.ru

Mukim Saidusainovich Saidusainov, Cand. Sci. (Phys.-Math.), Tajik National University, Dushanbe, 734025 Republic of Tajikistan,  e-mail: smuqim@gmail.com

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