论文标题
希尔伯特空间正交公式误差的下限
Lower Bounds for the Error of Quadrature Formulas for Hilbert Spaces
论文作者
论文摘要
我们证明了使用给定的样本点$ \ x_n = \ {x_1，\ dots，x_n \} $的最坏情况下的较低界限。我们主要对最佳点集$ \ x_n $感兴趣，但也证明了具有很高概率的独立和均匀分布点的概率的下限。作为一种工具，我们将Vybíral的最新结果（及其扩展）用于与Schur产品定理相关的某些矩阵的积极半定义。该新技术还适用于分析功能的空间，在无法应用基于可分解核的已知方法的空间。
We prove lower bounds for the worst case error of quadrature formulas that use given sample points $\X_n = \{ x_1, \dots , x_n \}$. We are mainly interested in optimal point sets $\X_n$, but also prove lower bounds that hold with high probability for sets of independently and uniformly distributed points. As a tool, we use a recent result (and extensions thereof) of Vybíral on the positive semi-definiteness of certain matrices related to the product theorem of Schur. The new technique also works for spaces of analytic functions where known methods based on decomposable kernels cannot be applied.
