Following recent interest by the community, the scaling of the minimal singular value of a Vandermonde matrix with nodes forming clusters on the length scale of Rayleigh distance on the complex unit circle is studied. Using approximation theoretic properties of exponential sums, we show that the decay is only single exponential in the size of the largest cluster, and the bound holds for arbitrary small minimal separation distance. We also obtain a generalization of well-known bounds on the smallest eigenvalue of the generalized prolate matrix in the multi-cluster geometry. Finally, the results are extended to the entire spectrum.

中文翻译：

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亚瑞利制度中范德蒙矩阵的最小奇异值的单指数界

根据社区最近的兴趣，研究了范德蒙矩阵的最小奇异值的缩放，其中节点在复单位圆上的瑞利距离的长度尺度上形成簇。使用指数和的近似理论性质，我们表明衰减只是最大集群大小的单指数，并且该界限适用于任意小的最小分离距离。我们还获得了多集群几何中广义长边矩阵的最小特征值的众所周知的界限的概括。最后，将结果扩展到整个频谱。