论文标题
零集的Grassmann几何形状在复制内核希尔伯特空间
Grassmann geometry of zero sets in reproducing kernel Hilbert spaces
论文作者
论文摘要
令$ \ Mathcal {H} $为集合$ x $上的复制品Hilbert函数空间。我们研究了找到$ \ Mathcal {H} $的Grassmann流形的最小测量的问题，该问题连接了两个子空间，这些子空间由$ X $的有限亚集成的功能消失。我们建立了一个必要且充分的条件，以实现地球学的存在和独特性，然后在示例中对其进行分析。当提到的有限子集为单例时，我们讨论了大地距离与其他已知指标的关系。我们发现对独特的自动化算子的上和下特征值的估计值定义了最小的大地测量学，当底层空间是强大的空间时，可以更精确。同样，对于强壮的空间，我们讨论了连接在磁盘的无限子集上的函数子空间的大地管的存在，并且我们调查了何时投影到这种类型的子空间上的乘积是紧凑的。
Let $\mathcal{H}$ be a reproducing kernel Hilbert space of functions on a set $X$. We study the problem of finding a minimal geodesic of the Grassmann manifold of $\mathcal{H}$ that joins two subspaces consisting of functions which vanish on given finite subsets of $X$. We establish a necessary and sufficient condition for existence and uniqueness of geodesics, and we then analyze it in examples. We discuss the relation of the geodesic distance with other known metrics when the mentioned finite subsets are singletons. We find estimates on the upper and lower eigenvalues of the unique self-adjoint operators which define the minimal geodesics, which can be made more precise when the underlying space is the Hardy space. Also for the Hardy space we discuss the existence of geodesics joining subspaces of functions vanishing on infinite subsets of the disk, and we investigate when the product of projections onto this type of subspaces is compact.
