亚Rayleigh区域下Vandermonde矩阵最小奇异值的单指数界
Single-exponential bounds for the smallest singular value of Vandermonde matrices in the sub-Rayleigh regime
摘要：
根据最近的兴趣，社区，范德蒙矩阵的最小奇异值的缩放与节点形成集群上的瑞利距离的长度尺度上的复杂单位圆的研究。利用指数和的近似理论性质，证明了在最大簇的尺度下，衰减仅为单指数衰减，并且对于任意小的最小分离距离，该界成立.我们还得到了多簇几何中广义长矩矩阵最小特征值的一个推广。最后，将结果推广到整个谱。
英文摘要：
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.
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