Optimal recovery of operators in sequence spaces

Abstract

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.