Optimal recovery of operator sequences

Abstract

In this paper we consider two recovery problems based on information given with an error. First is the problem of optimal recovery of the class $W^T_q = \{(t_1h_1,t_2h_2,\ldots)\in \ell_q\,:\,\|h\|_q\leqslant 1\}$, where $1\le q < \infty$ and $t_1\geqslant t_2\geqslant \ldots \geqslant 0$, in the space $\ell_q$ when in the capacity of inexact information we know either the first $n\in\mathbb{N}$ elements of a sequence with an error measured in the space of finite sequences $\ell_r^n$, $0 < r \le \infty$, or a sequence itself is known with an error measured in the space $\ell_r$. The second is the problem of optimal recovery of scalar products acting on 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 $s_1\ge s_2\ge \ldots \ge 0$, when in the capacity of inexact information we know the first $n$ coordinate-wise products $x_1y_1, x_2y_2,\ldots,x_ny_m$ of the element $x\times y \in W^{T,S}_{p,q}$ with an error measured in the space $\ell_r^n$. We find exact solutions to above problems and construct optimal methods of recovery. As an application of our results we consider the problem of optimal recovery of classes in Hilbert spaces by Fourier coefficients known with an error measured in the space $\ell_p$ with $p > 2$.