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Convergence of remote projections onto convex setsстатья
- Авторы: Borodin Petr A., Kopecka Eva
- Журнал: Pure and Applied Functional Analysis
- Том: 8
- Номер: 6
- Год издания: 2023
- Первая страница: 1603
- Последняя страница: 1620
- Аннотация: Let {Cα}α∈Ω be a family of closed and convex sets in a Hilbert space H, having a nonempty intersection C. We consider a sequence {xn} of remote projections onto them. This means, x0∈H, and xn+1 is the projection of xn onto such a set Cα(n) that the ratio of the distances from xn to this set and to any other set from the family is at least tn∈[0,1]. We study properties of the weakness parameters tn and of the sets Cα which ensure the norm or weak convergence of the sequence {xn} to a point in C. We show that condition (T) is necessary and sufficient for the norm convergence of xn to a point in C for any starting element and any family of closed, convex, and symmetric sets Cα. 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 Cα can be dropped. Condition (T) is stronger than ∑t2n=∞ and weaker than ∑tn/n=∞. The condition ∑t2n=∞ turns out to be necessary and sufficient for the sequence {xn} to have a partial weak limit in C for any family of closed and convex sets Cα and any starting element.
- Добавил в систему: Бородин Петр Анатольевич