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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | https://arxiv.org/abs/2312.17574 |
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| _version_ | 1866916076576571392 |
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| author | Borodin, Petr A. Kopecká, Eva |
| author_facet | Borodin, Petr A. Kopecká, Eva |
| contents | Let $\{C_α\}_{α\in Ω}$ be a family of closed and convex sets in a Hilbert space $H$, having a nonempty intersection $C$. We consider a sequence $\{x_n\}$ of remote projections onto them. This means, $x_0\in H$, and $x_{n+1}$ is the projection of $x_n$ onto such a set $C_{α(n)}$ that the ratio of the distances from $x_n$ to this set and to any other set from the family is at least $t_n\in [0,1]$. We study properties of the weakness parameters $t_n$ and of the sets $C_α$ which ensure the norm or weak convergence of the sequence $\{x_n\}$ to a point in $C$. We show that condition (T) is necessary and sufficient for the norm convergence of $x_n$ to a point in $C$ for any starting element and any family of closed, convex, and symmetric sets $C_α$. This generalizes a result of Temlyakov who introduced (T) in the context of greedy approximation theory. We give examples explaining to what extent the symmetry condition on the sets $C_α$ can be dropped. Condition (T) is stronger than $\sum t_n^2=\infty$ and weaker than $\sum t_n/n=\infty$. The condition $\sum t_n^2=\infty$ turns out to be necessary and sufficient for the sequence $\{x_n\}$ to have a partial weak limit in $C$ for any family of closed and convex sets $C_α$ and any starting element. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_17574 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Convergence of remote projections onto convex sets Borodin, Petr A. Kopecká, Eva Functional Analysis 46C05, 47J25 Let $\{C_α\}_{α\in Ω}$ be a family of closed and convex sets in a Hilbert space $H$, having a nonempty intersection $C$. We consider a sequence $\{x_n\}$ of remote projections onto them. This means, $x_0\in H$, and $x_{n+1}$ is the projection of $x_n$ onto such a set $C_{α(n)}$ that the ratio of the distances from $x_n$ to this set and to any other set from the family is at least $t_n\in [0,1]$. We study properties of the weakness parameters $t_n$ and of the sets $C_α$ which ensure the norm or weak convergence of the sequence $\{x_n\}$ to a point in $C$. We show that condition (T) is necessary and sufficient for the norm convergence of $x_n$ to a point in $C$ for any starting element and any family of closed, convex, and symmetric sets $C_α$. This generalizes a result of Temlyakov who introduced (T) in the context of greedy approximation theory. We give examples explaining to what extent the symmetry condition on the sets $C_α$ can be dropped. Condition (T) is stronger than $\sum t_n^2=\infty$ and weaker than $\sum t_n/n=\infty$. The condition $\sum t_n^2=\infty$ turns out to be necessary and sufficient for the sequence $\{x_n\}$ to have a partial weak limit in $C$ for any family of closed and convex sets $C_α$ and any starting element. |
| title | Convergence of remote projections onto convex sets |
| topic | Functional Analysis 46C05, 47J25 |
| url | https://arxiv.org/abs/2312.17574 |