非均匀傅里叶矩阵的最小奇异值
Smallest Singular Value Estimates for Nonuniform Fourier Matrices via Periodic Nonuniform Sampling
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中文总结 AI 辅助
该研究在节点聚类和等距网格扰动两种场景下，将非均匀傅里叶矩阵最小奇异值问题归约为周期非均匀插值矩阵的谱范数估计，得到接近最优界，证实了Austin与Trefethen关于2-范数勒贝格常数的部分猜想。
中文摘要 AI 辅助
我们在两种场景下研究非均匀傅里叶矩阵的最小奇异值：节点聚类和等距网格的扰动。通过将该问题归约为周期非均匀插值矩阵的谱范数估计，我们在两种场景下均得到了接近最优的界。对于聚类节点，我们推导了一个局部分离条件，其中每个所需的间隙仅取决于两个相邻聚类的大小。对于界为1/4≤L<1/2的扰动，我们的结果在对数因子范围内证实了Austin和Trefethen关于2-范数勒贝格常数的猜想。
英文摘要
We study the smallest singular value of nonuniform Fourier matrices in two settings: clustered nodes and perturbations of an equispaced grid. By reducing the problem to spectral norm estimates for periodic nonuniform interpolation matrices, we obtain nearly optimal bounds in both cases. For clustered nodes, we derive the first local separation condition in which each required gap depends only on the sizes of the two neighboring clusters. For perturbations with the bound \(1/4\leq L<1/2\), our result confirms the conjecture of Austin and Trefethen on the \(2\)-norm Lebesgue constant up to a logarithmic factor.
