On the singular value distribution of Vandermonde matrix with nodes on the unit circle

Question

I want to know about the spectral property of the following Vandermonde matrix,

\begin{align} \begin{bmatrix} z^{0\times 0} & z^{0\times 1} & \cdots & z^{0\times (N-1)}\\ z^{1\times 0} & z^{1\times 1} & \cdots & z^{1\times (N-1)}\\ \vdots & \vdots & \ddots & \vdots\\ z^{(N-1)\times 0} & z^{(N-1)\times 1} & \cdots & z^{(N-1)\times (N-1)}\\ \end{bmatrix} \end{align}

where $z$ is a complex number on the unit circle. Two obvious extreme cases, $z=1$ and $z=e^{-2\pi i/N}$, give rank-1 and scaled-unitary matrix, respectively. Hence, the first case offers the single nonzero singular value of $N$ and the second case offers $N$ equal singular values of $\sqrt{N}$. I found that, for general $z$ on the unit circle, most of singular values are clustered at both ends, almost zero or very big for large $N$. Is there any literature explaining this asymptotic result?

Thanks.

Answer 1

Here are some references regarding the singular values of a Vandermonde matrix with nodes on the unit circle:

[1] Dmitry Batenkov, Benedikt Diederichs, Gil Goldman, and Yosef Yomdin, "The spectral properties of Vandermonde matrices with clustered nodes," https://arxiv.org/abs/1909.01927

[2] Weilin Li and Wenjing Liao, "Stable Super-Resolution Limit and Smallest Singular Value of Restricted Fourier Matrices," Applied and Computational Harmonic Analysis, 2021

[3] Benedikt Diederichs, "Well-Posedness of Sparse Frequency Estimation," https://arxiv.org/abs/1905.08005

[4] Gabriel H. Tucci and Philip A. Whiting, "The Spectral Properties of Vandermonde Matrices with Clustered Nodes," https://arxiv.org/abs/1909.01927