Optimal recovery of operator sequences

Abstract

In this paper we solve two problems of optimal recovery based on information given with an error. First is the problem of optimal recovery of the class $W^T_q = \{(t_1h_1,t_2h_2,\ldots)\,\colon \,\|h\|_{\ell_q}\le 1\}$, where $1\le q < \infty$ and $t_1\ge t_2\ge \ldots \ge 0$ are given, in the space $\ell_q$. 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 $\ell_r^n$, $0 < r \le \infty$, does not exceed given $\varepsilon\ge 0$, or (ii) by a sequence $y\in\ell_\infty$ 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 $y\in\ell^n_r$), where

\smallskip\centerline{$\displaystyle
\Phi^*_m(y) = \Big\{y_1\left(1 - \frac{t_{m+1}^q}{t_{1}^q}\Big),\ldots,y_m\Big(1 - \frac{t_{m+1}^q}{t_{m}^q}\Big),0,\ldots\right\},\quad y\in\mathbb{R}^n\text{ or } y\in\ell_\infty,$}

\smallskip\noi
or convex combination $(1-\lambda) \Phi^*_{m+1} + \lambda\Phi^*_{m}$.

The second one is the problem of optimal recovery of the scalar product operator acting on the Cartesian product $W^{T,S}_{p,q}$ of classes $W^T_p$ and $W^S_q$, where $1 < p,q < \infty$, $\frac{1}{p} + \frac{1}{q} = 1$ and $s_1\ge s_2\ge \ldots \ge 0$ are given. Information available about elements $x\in W^T_p$ and $y\in W^S_q$ is provided by elements $z,w\in \mathbb{R}^n$ such that the distance between vectors $\left(x_1y_1, x_2y_2,\ldots,x_ny_n\right)$ and $\left(z_1w_1,\ldots,z_nw_n\right)$ in the space $\ell_r^n$ does not exceed $\varepsilon$. We show that the optimal method of recovery is delivered either by operator $\Psi^*_m$ with some $m\in\{0,1,\ldots,n\}$, where

\smallskip\centerline{$\displaystyle
\Psi^*_m = \sum_{k=1}^m z_kw_k\Big(1 - \frac{t_{m+1}s_{m+1}}{t_ks_k}\Big),\quad z,w\in\mathbb{R}^n,$}

\smallskip\noi
or by convex combination $(1-\lambda)\Psi^*_{m+1} + \lambda\Psi^*_{m}$.

As an application of our results we consider the problem of optimal recovery of classes in Hilbert spaces by the Fourier coefficients of its elements known with an error measured in the space $\ell_p$ with $p > 2$.