将Meshulam关于松弛投影到仿射子空间轨道有界性的结果从有限维扩展到无限维Hilbert空间
Extending Meshulam's result on the boundedness of orbits of relaxed projections onto affine subspaces from finite to infinite-dimensional Hilbert spaces
AI总结：
本文通过归纳子空间数量，将Meshulam关于有限维欧氏空间中随机投影到仿射子空间序列有界性的结果推广到一般Hilbert空间，要求平行线性子空间具有内在正则性，并讨论了结果的精确性及其与随机块Kaczmarz方法的联系。
摘要：
1996年，Meshulam证明了欧氏空间中从有限集合中随机投影到仿射子空间生成的任何序列即使子空间交集为空也保持有界。他的证明甚至适用于松弛投影，依赖于对欧氏空间维数的巧妙归纳。在本文中，我们通过对给定集合中仿射子空间数量的归纳证明，将Meshulam的结果推广到一般Hilbert空间设置。我们要求相应的平行线性子空间是内在正则的——这一假设在欧氏空间中总是成立。我们还讨论了结果的精确性，并建立了与随机块Kaczmarz方法的联系。
英文摘要：
In 1996, Meshulam proved that any sequence generated in Euclidean space by randomly projecting onto affine subspaces drawn from a finite collection stays bounded even if the intersection of the subspaces is empty. His proof, which works even for relaxed projections, relies on an ingenious induction on the dimension of the Euclidean space. In this paper, we extend Meshulam's result to the general Hilbert space setting by an induction proof of the number of affine subspaces in the given collection. We require that the corresponding parallel linear subspaces are innately regular -- this assumption always holds in Euclidean space. We also discuss the sharpness of our result and make a connection to randomized block Kaczmarz methods.
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