论文标题
希尔伯特空间中的最佳近似映射
Best approximation mappings in Hilbert spaces
论文作者
论文摘要
Behling，Bello Cruz和Santos引入了相对于有限维空间中封闭的仿射子空间的最佳近似映射（BAM）的概念，以显示块状周围反射方法的线性收敛。最佳的近似映射具有线性收敛的圆周映射的两个临界属性。 由于BAM的迭代顺序线性收敛，因此BAM本身就是有趣的。在本文中，我们自然地将BAM的定义从封闭的Aggine子空间扩展到非空的封闭凸集，从$ \ Mathbb {r}^{n} $到Hilbert Space。我们发现与BAM关联的凸集必须是BAM的固定点集。因此，由BAM生成的迭代序列线性收敛到BAM的最近固定点。 BAMS与其他映射之间产生收敛迭代序列之间的连接。 Behling等人证明了与封闭的仿射子空间相关的BAMS的有限组成仍然是$ \ Mathbb {r}^{n} $中的BAM。我们将它们的结果从$ \ mathbb {r}^{n} $概括为Hilbert Space，还构建了与BAMS组成相关的新常数。这提供了交替投影方法线性收敛的新证明。此外，研究了与一般凸组相关的BAM的组成。此外，我们表明与仿射子空间相关的BAMS的凸组合是BAM。最后但并非最不重要的一点是，我们将BAM与希尔伯特空间中的圆周映射联系起来。
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.
