Optimal recovery of operators in sequence spaces - V.F. Babenko; N.V. Parfinovych; D.S. Skorokhodov (Researches in Mathematics)
In this paper we solve the problem of optimal recovery of the operator $A_α x= (α_1x_1,α_2x_2,…)$ on the class $W^T_q = \{(t_1h_1,t_2h_2,…)\,:\,\|h\|_{\ell_q}\le 1\}$, where $1\le q < ∞$ and $t_1\ge t_2\ge … \ge 0$, and $α_1t_1\geα_2t_2\ge…\ge 0$ are given, in the space $\ell_q$. We solve this problem under assumption that $\lim_{n→∞}t_n = \lim_{n→∞}α_nt_n = 0$. 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 $\left(x_1,…,x_n\right)$ of $x$ in the space $\ell_p^n$, $0 < p \le ∞$, does not exceed given $ε\ge 0$, or (ii) by a sequence $y\in\ell_p$ whose distance to $x$ in the space $\ell_r$ does not exceed $ε$. We show that the optimal method of recovery in this problem is either operator $Φ^*_m$ with some $m\in\mathbb{Z}_+$ ($m\le n$ in case $y\in\ell^n_p$), defined by $$ Φ^*_m(y) = \left\{α_1y_1\left(1 - \frac{α_{m+1}^qt_{m+1}^q}{α_1^qt_{1}^q}\right),…,α_my_m\left(1 - \frac{α_{m+1}^qt_{m+1}^q}{α_m^qt_{m}^q}\right),0,…\right\}, $$ where $y\in\mathbb{R}^n$ or $y\in\ell_p$ or convex combination $(1-λ) Φ^*_{m+1} + λΦ^*_{m}$, or the operator $A_α$ itself.