论文标题
关于光谱测试与各向同性差异的关系和$ L_Q $ -Approximation在Sobolev空间中
On the relation of the spectral test to isotropic discrepancy and $L_q$-approximation in Sobolev spaces
论文作者
论文摘要
本文是最近论文“有关晶格点集的各向同性差异和光谱测试的注释” [J.复杂性，58：101441，2020]。我们表明，晶格点集的各向同性差异最多是$ d \，2^{2（d+1）} $乘以其光谱测试，从而纠正了对尺寸$ d $的依赖性，并且在提到的论文中的上限2中的上限证明了对尺寸的依赖性。主要任务是绑定单位立方体中包含的凸组集合的邻域的体积。 此外，我们表征了在光谱测试方面的距离距离设置的平均值。作为一种应用，我们推断出光谱测试 - 以及各向同性差异 - 对于晶格点设定的Sobolev函数近似的适合性至关重要。
This paper is a follow-up to the recent paper "A note on isotropic discrepancy and spectral test of lattice point sets" [J. Complexity, 58:101441, 2020]. We show that the isotropic discrepancy of a lattice point set is at most $d \, 2^{2(d+1)}$ times its spectral test, thereby correcting the dependence on the dimension $d$ and an inaccuracy in the proof of the upper bound in Theorem 2 of the mentioned paper. The major task is to bound the volume of the neighbourhood of the boundary of a convex set contained in the unit cube. Further, we characterize averages of the distance to a lattice point set in terms of the spectral test. As an application, we infer that the spectral test -- and with it the isotropic discrepancy -- is crucial for the suitability of the lattice point set for the approximation of Sobolev functions.
