We prove lower bounds for the worst case error of quadrature formulas that use given sample points ${\mathcal{X}}_n=\{x_1,\dots ,x_n\}$. We are mainly interested in optimal point sets ${\mathcal{X}}_n$, 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.

中文翻译：

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Hilbert空间的正交公式的误差的下界

我们证明了使用给定采样点的正交公式的最坏情况错误的下界 ${\mathcal{X}}_{ñ}=\{X_{\mathrm{1个}}，\dots ，X_{ñ}\}$。我们主要对最佳点集感兴趣${\mathcal{X}}_{ñ}$，也证明了下界对于独立且均匀分布的点集具有较高的概率。作为一种工具，我们使用Vybíral的最新结果（及其扩展），该结果关于与Schur乘积定理有关的某些矩阵的正半定性。这项新技术还适用于无法使用基于可分解内核的已知方法的解析函数空间。