Optimal recovery of operators in sequence spaces

In this paper we solve the problem of optimal recovery of the operator $A_\alpha x= (\alpha_1x_1,\alpha_2x_2,\ldots)$ on the class $W^T_q = \{(t_1h_1,t_2h_2,\ldots)\,:\,\|h\|_{\ell_q}\le 1\}$, where $1\le q < \infty$ and $t_1\ge t_2\ge \ldots \ge 0$, and $\alpha_1t_1\ge\alpha_2t_2\ge\ldots\ge 0$ are given, in the space $\ell_q$. We solve this problem under assumption that $\lim_{n\to\infty}t_n = \lim_{n\to\infty}\alpha_nt_n = 0$. Information available about a sequence $x\in W^T_q$ is provided either (i) by an element $y\in\mathbb{R}^n$, $n\in\mathbb{N}$, whose distance to the first $n$ coordinates $\left(x_1,\ldots,x_n\right)$ of $x$ in the space $\ell_p^n$, $0 < p \le \infty$, does not exceed given $\varepsilon\ge 0$, or (ii) by a sequence $y\in\ell_p$ whose distance to $x$ in the space $\ell_r$ does not exceed $\varepsilon$. We show that the optimal method of recovery in this problem is either operator $\Phi^*_m$ with some $m\in\mathbb{Z}_+$ ($m\le n$ in case $y\in\ell^n_p$), defined by $$\Phi^*_m(y) = \left\{\alpha_1y_1\left(1 - \frac{\alpha_{m+1}^qt_{m+1}^q}{\alpha_1^qt_{1}^q}\right),\ldots,\alpha_my_m\left(1 - \frac{\alpha_{m+1}^qt_{m+1}^q}{\alpha_m^qt_{m}^q}\right),0,\ldots\right\},$$where $y\in\mathbb{R}^n$ or $y\in\ell_p$ or convex combination $(1-\lambda) \Phi^*_{m+1} + \lambda\Phi^*_{m}$, or the operator $A_\alpha$ itself.