Optimal recovery of operator sequences

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