This paper studies the extreme singular values of non-harmonic Fourier matrices. Such a matrix 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 $\mathcal{X}=\{x_k\}_{k=1}^s$ and $m\geq s$. A main result provides an explicit lower bound for the smallest singular value of $\Phi$ under the assumption $m\geq 6s$ and without any restrictions on $\mathcal{X}$. It shows 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 result significantly improves upon classical bounds and achieves 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。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 $\mathcal{X}=\{x_k\}_{k=1}^s$ 且 $m\geq s$。主要结果给出了在假设 $m\geq 6s$ 且不对 $\mathcal{X}$ 施加任何限制的条件下，$\Phi$ 最小奇异值的显式下界。结果表明：对于由密度准则确定的合适尺度 $\tau$，$\mathcal{X}$ 中元素在小于 $\tau$ 的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 $\tau$ 的距离则不太重要，仅取决于远距离点的局部稀疏性。理论与数值比较表明，该主要结果显著改进了经典界限，并达到了先前在更具限制性条件下发现的相同速率。