AI 中文总结
本研究在鲍施克和董的基础上，通过允许扰动、考虑闭半空间与超平面混合及使用动态加权和等方式扩展其关于松弛投影无穷乘积有界性的结果，还讨论推广局限并建立赋范空间压缩映射无穷乘积相关定理。
AI 中文摘要
最近（2026年），鲍施克和董将梅舒拉姆1996年发表的一个结果（继1984年阿哈罗尼 - 迪谢 - 瓦伊恩里布的早期结果之后）从有限维希尔伯特空间扩展到了无限维，该结果涉及到在有限个闭仿射子空间上松弛投影的无穷乘积的有界性。在本笔记中，我们通过多种方式扩展了鲍施克和董的结果，包括允许某些扰动并证明扰动弹性，考虑闭半空间和闭超平面的混合情况，以及在迭代过程中使用松弛投影的动态字符串（动态长度）的动态加权和。我们还通过给出一大类反例来讨论将鲍施克 - 董结果推广到任意闭凸集的局限性，其中相关控制不是循环的甚至不是几乎循环的。在此过程中，我们建立了一个关于赋范空间中压缩映射无穷乘积的一致有界性和一致无界性的具有独立意义的一般定理。
英文摘要
Very recently (2026), Bauschke and Tung extended from finite- to infinite-dimensional Hilbert spaces a result published by Meshulam in 1996 (following an earlier result of Aharoni-Duchet-Wajnryb from 1984) regarding the boundedness of infinite products of relaxed projections onto a finite family of closed affine subspaces. In the present note we extend in various ways the result of Bauschke and Tung by allowing certain perturbations and proving perturbation resilience, by considering a mixture of closed half-spaces and closed hyperplanes, and by using dynamic weighted sums of dynamic strings (of dynamic lengths) of relaxed projections in the iterative process. We also discuss the limitation to generalize the Bauschke-Tung result to arbitrary closed and convex sets by presenting a large family of counterexamples in which the associated control is not cyclic and not even almost cyclic. Along the way we establish a general theorem of independent interest regarding the uniform boundedness and uniform unboundedness of infinite products of nonexpansive mappings in a normed space setting.
