Optimal recovery of operator sequences
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 sp...
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Ivan Franko National University of Lviv
2021-12-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/257 |
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| author | V. F. Babenko N. V. Parfinovych D. S. Skorokhodov |
| author_facet | V. F. Babenko N. V. Parfinovych D. S. Skorokhodov |
| author_sort | V. F. Babenko |
| collection | DOAJ |
| description | 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 $p > 2$. |
| format | Article |
| id | doaj-art-cf954facec9f48ffa1b5da9d5fc81dea |
| institution | Kabale University |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2021-12-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-cf954facec9f48ffa1b5da9d5fc81dea2025-08-20T03:33:11ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202021-12-0156219320710.30970/ms.56.2.193-207257Optimal recovery of operator sequencesV. F. Babenko0N. V. Parfinovych1D. S. Skorokhodov2Mechanics and Mathematics Department, Oles Honchar Dnipro National University Dnipro, UkraineMechanics and Mathematics Department Oles Honchar Dnipro National University Dnipro, UkraineMechanics and Mathematics Department Oles Honchar Dnipro National University Dnipro, UkraineIn 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 $p > 2$.http://matstud.org.ua/ojs/index.php/matstud/article/view/257optimal recovery of operatorsmethod of recoveryrecovery with non-exact informationsequence spaces |
| spellingShingle | V. F. Babenko N. V. Parfinovych D. S. Skorokhodov Optimal recovery of operator sequences Математичні Студії optimal recovery of operators method of recovery recovery with non-exact information sequence spaces |
| title | Optimal recovery of operator sequences |
| title_full | Optimal recovery of operator sequences |
| title_fullStr | Optimal recovery of operator sequences |
| title_full_unstemmed | Optimal recovery of operator sequences |
| title_short | Optimal recovery of operator sequences |
| title_sort | optimal recovery of operator sequences |
| topic | optimal recovery of operators method of recovery recovery with non-exact information sequence spaces |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/257 |
| work_keys_str_mv | AT vfbabenko optimalrecoveryofoperatorsequences AT nvparfinovych optimalrecoveryofoperatorsequences AT dsskorokhodov optimalrecoveryofoperatorsequences |