This paper studies the extreme singular values of non-harmonic Fourier matrices. Such a matrix of size $m\times s$ can be written as $\Phi=[ e^{-2\pi i j x_k}]_{j=0,1,\dots,m-1, k=1,2,\dots,s}$ for some set $\mathcal{X}=\{x_k\}_{k=1}^s$. Its condition number controls the stability of inversion, which is of great importance to super-resolution and nonuniform Fourier transforms. Under the assumption $m\geq 6s$ and without any restrictions on $\mathcal{X}$, the main theorems provide explicit lower bounds for the smallest singular value $\sigma_s(\Phi)$ in terms of distances between elements in $\mathcal{X}$. More specifically, distances exceeding an appropriate scale $\tau$ have modest influence on $\sigma_s(\Phi)$, while the product of distances that are less than $\tau$ dominates the behavior of $\sigma_s(\Phi)$. These estimates reveal how the multiscale structure of $\mathcal{X}$ affects the condition number of Fourier matrices. Theoretical and numerical comparisons indicate that the main theorems significantly improve upon classical bounds and recover the same rate for special cases but with relaxed assumptions.

翻译：本文研究非调和傅里叶矩阵的极端奇异值。此类大小为 $m\times s$ 的矩阵可表示为 $\Phi=[ e^{-2\pi i j x_k}]_{j=0,1,\dots,m-1, k=1,2,\dots,s}$，其中 $\mathcal{X}=\{x_k\}_{k=1}^s$ 为某个点集。其条件数控制着求逆的稳定性，这对于超分辨率和非均匀傅里叶变换至关重要。在假设 $m\geq 6s$ 且对 $\mathcal{X}$ 不作任何限制的条件下，主要定理依据 $\mathcal{X}$ 中元素之间的距离，为最小奇异值 $\sigma_s(\Phi)$ 提供了显式的下界。具体而言，超过适当尺度 $\tau$ 的距离对 $\sigma_s(\Phi)$ 的影响有限，而小于 $\tau$ 的距离之积主导了 $\sigma_s(\Phi)$ 的行为。这些估计揭示了 $\mathcal{X}$ 的多尺度结构如何影响傅里叶矩阵的条件数。理论与数值比较表明，主要定理相较于经典界有显著改进，并在放宽假设的条件下，对特殊情形恢复了相同的收敛速率。