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 $\mathcal{X}=\{x_k\}_{k=1}^s$. The main results provide explicit lower bounds for the smallest singular value of $\Phi$ under the assumption $m\geq 6s$ and without any restrictions on $\mathcal{X}$. They show that for an appropriate scale $\tau$ determined by a density criteria, interactions between elements in $\mathcal{X}$ at scales smaller than $\tau$ are most significant and depends on the multiscale structure of $\mathcal{X}$ at fine scales, while distances larger than $\tau$ are less important and only depend on the local sparsity of the far away points. Theoretical and numerical comparisons show that the main results significantly improve upon classical bounds and achieve the same rate that was previously discovered for more restrictive settings.

翻译：本文研究非调和傅里叶矩阵的极端奇异值。对于大小为 $m\times s$ 的此类矩阵，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 $\mathcal{X}=\{x_k\}_{k=1}^s$ 为某集合。主要结果在假设 $m\geq 6s$ 且不对 $\mathcal{X}$ 施加任何限制的条件下，给出了 $\Phi$ 最小奇异值的显式下界。结果表明，对于由密度准则确定的适当尺度 $\tau$，$\mathcal{X}$ 中元素在小于 $\tau$ 尺度上的相互作用最为显著，且依赖于 $\mathcal{X}$ 在精细尺度上的多尺度结构；而大于 $\tau$ 的距离则重要性较低，仅取决于远距离点的局部稀疏性。理论与数值比较显示，主要结果显著优于经典界，并达到了先前仅在更严格条件下发现的相同速率。