Optimal recovery of operator sequences
Анотація
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 WTq={(t1h1,t2h2,…):‖, 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.
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