希尔伯特空间中的最佳逼近映射
Best approximation mappings in Hilbert spaces
摘要：
Behling，Bello Cruz和Santos在有限维空间中引入了关于闭仿射子空间的最佳逼近映射（BAM）的概念，以证明分块包围反射法的线性收敛.最佳逼近映射具有外心映射线性收敛的两个重要性质。由于BAM的迭代序列线性收敛，BAM本身就很有趣。本文将BAM的定义从闭仿射子空间自然地推广到非空闭凸集，从$\mathbb{R}^{n}$自然地推广到一般希尔伯特空间。我们发现与BAM相关联的凸集必定是BAM的不动点集。因此，由BAM生成的迭代序列线性地收敛到BAM的最近的不动点。考虑了BAM与其他生成收敛迭代序列的映射之间的联系。Behling等人证明了与闭仿射子空间相关联的BAM的有限合成仍然是$\mathbb{R}^{n}$中的BAM。我们将他们的结果从$\mathbb{R}^{n}$推广到一般的希尔伯特空间，并构造了一个新的与BAM的合成有关的常数.这为交替投影法的线性收敛提供了一个新的证明。此外，与一般凸集的BAM的合成进行了研究。此外，我们证明了与仿射子空间相关的BAM的凸组合是BAM。最后，我们将BAM与希尔伯特空间中的外心映射联系起来。
英文摘要：
The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional space was introduced by Behling, Bello Cruz and Santos to show the linear convergence of the block-wise circumcentered-reflection method. The best approximation mapping possesses two critical properties of the circumcenter mapping for linear convergence. Because the iteration sequence of BAM linearly converges, the BAM is interesting in its own right. In this paper, we naturally extend the definition of BAM from closed affine subspace to nonempty closed convex set and from $\mathbb{R}^{n}$ to general Hilbert space. We discover that the convex set associated with the BAM must be the fixed point set of the BAM. Hence, the iteration sequence generated by a BAM linearly converges to the nearest fixed point of the BAM. Connections between BAMs and other mappings generating convergent iteration sequences are considered. Behling et al.\ proved that the finite composition of BAMs associated with closed affine subspaces is still a BAM in $\mathbb{R}^{n}$. We generalize their result from $\mathbb{R}^{n}$ to general Hilbert space and also construct a new constant associated with the composition of BAMs. This provides a new proof of the linear convergence of the method of alternating projections. Moreover, compositions of BAMs associated with general convex sets are investigated. In addition, we show that convex combinations of BAMs associated with affine subspaces are BAMs. Last but not least, we connect BAM with circumcenter mapping in Hilbert spaces.
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