Optimal recovery of operators in sequence spaces
In this paper we solve the problem of optimal recovery of the operator $A_\alpha x= (\alpha_1x_1,\alpha_2x_2,\ldots)$ on the class $W^T_q = \{(t_1h_1,t_2h_2,\ldots)\,:\,\|h\|_{\ell_q}\le 1\}$, where $1\le q < \infty$ and $t_1\ge t_2\ge \ldots \ge 0$, and $\alpha_1t_1\ge\alpha_2t_2\ge\ldots\ge 0$...
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Oles Honchar Dnipro National University
2024-12-01
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| Series: | Researches in Mathematics |
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| Online Access: | https://vestnmath.dnu.dp.ua/index.php/rim/article/view/431/431 |
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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 the problem of optimal recovery of the operator $A_\alpha x= (\alpha_1x_1,\alpha_2x_2,\ldots)$ on the class $W^T_q = \{(t_1h_1,t_2h_2,\ldots)\,:\,\|h\|_{\ell_q}\le 1\}$, where $1\le q < \infty$ and $t_1\ge t_2\ge \ldots \ge 0$, and $\alpha_1t_1\ge\alpha_2t_2\ge\ldots\ge 0$ are given, in the space $\ell_q$. We solve this problem under assumption that $\lim_{n\to\infty}t_n = \lim_{n\to\infty}\alpha_nt_n = 0$. 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 $\ell_p^n$, $0 < p \le \infty$, does not exceed given $\varepsilon\ge 0$, or (ii) by a sequence $y\in\ell_p$ 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 $y\in\ell^n_p$), defined by
$$
\Phi^*_m(y) = \left\{\alpha_1y_1\left(1 - \frac{\alpha_{m+1}^qt_{m+1}^q}{\alpha_1^qt_{1}^q}\right),\ldots,\alpha_my_m\left(1 - \frac{\alpha_{m+1}^qt_{m+1}^q}{\alpha_m^qt_{m}^q}\right),0,\ldots\right\},
$$
where $y\in\mathbb{R}^n$ or $y\in\ell_p$ or convex combination $(1-\lambda) \Phi^*_{m+1} + \lambda\Phi^*_{m}$, or the operator $A_\alpha$ itself. |
| format | Article |
| id | doaj-art-9dff053a8cb447de8c4549590126690f |
| institution | Kabale University |
| issn | 2664-4991 2664-5009 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Oles Honchar Dnipro National University |
| record_format | Article |
| series | Researches in Mathematics |
| spelling | doaj-art-9dff053a8cb447de8c4549590126690f2025-01-05T19:28:20ZengOles Honchar Dnipro National UniversityResearches in Mathematics2664-49912664-50092024-12-01322405210.15421/242418Optimal recovery of operators in sequence spacesV.F. Babenko0https://orcid.org/0000-0001-6677-1914N.V. Parfinovych1https://orcid.org/0000-0002-3448-3798D.S. Skorokhodov2https://orcid.org/0000-0001-8494-5885Oles Honchar Dnipro National UniversityOles Honchar Dnipro National UniversityOles Honchar Dnipro National UniversityIn this paper we solve the problem of optimal recovery of the operator $A_\alpha x= (\alpha_1x_1,\alpha_2x_2,\ldots)$ on the class $W^T_q = \{(t_1h_1,t_2h_2,\ldots)\,:\,\|h\|_{\ell_q}\le 1\}$, where $1\le q < \infty$ and $t_1\ge t_2\ge \ldots \ge 0$, and $\alpha_1t_1\ge\alpha_2t_2\ge\ldots\ge 0$ are given, in the space $\ell_q$. We solve this problem under assumption that $\lim_{n\to\infty}t_n = \lim_{n\to\infty}\alpha_nt_n = 0$. 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 $\ell_p^n$, $0 < p \le \infty$, does not exceed given $\varepsilon\ge 0$, or (ii) by a sequence $y\in\ell_p$ 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 $y\in\ell^n_p$), defined by $$ \Phi^*_m(y) = \left\{\alpha_1y_1\left(1 - \frac{\alpha_{m+1}^qt_{m+1}^q}{\alpha_1^qt_{1}^q}\right),\ldots,\alpha_my_m\left(1 - \frac{\alpha_{m+1}^qt_{m+1}^q}{\alpha_m^qt_{m}^q}\right),0,\ldots\right\}, $$ where $y\in\mathbb{R}^n$ or $y\in\ell_p$ or convex combination $(1-\lambda) \Phi^*_{m+1} + \lambda\Phi^*_{m}$, or the operator $A_\alpha$ itself.https://vestnmath.dnu.dp.ua/index.php/rim/article/view/431/431optimal recovery of operatorsmethod of recoveryrecovery with non-exact informationsequence spaces |
| spellingShingle | V.F. Babenko N.V. Parfinovych D.S. Skorokhodov Optimal recovery of operators in sequence spaces Researches in Mathematics optimal recovery of operators method of recovery recovery with non-exact information sequence spaces |
| title | Optimal recovery of operators in sequence spaces |
| title_full | Optimal recovery of operators in sequence spaces |
| title_fullStr | Optimal recovery of operators in sequence spaces |
| title_full_unstemmed | Optimal recovery of operators in sequence spaces |
| title_short | Optimal recovery of operators in sequence spaces |
| title_sort | optimal recovery of operators in sequence spaces |
| topic | optimal recovery of operators method of recovery recovery with non-exact information sequence spaces |
| url | https://vestnmath.dnu.dp.ua/index.php/rim/article/view/431/431 |
| work_keys_str_mv | AT vfbabenko optimalrecoveryofoperatorsinsequencespaces AT nvparfinovych optimalrecoveryofoperatorsinsequencespaces AT dsskorokhodov optimalrecoveryofoperatorsinsequencespaces |