Consider an operator that takes the Fourier transform of a discrete measure supported in $\mathcal{X}\subset[-\frac 12,\frac 12)^d$ and restricts it to a compact $\Omega\subset\mathbb{R}^d$. We provide lower bounds for its smallest singular value when $\Omega$ is either a closed ball of radius $m$ or closed cube of side length $2m$, and under different types of geometric assumptions on $\mathcal{X}$. We first show that if distances between points in $\mathcal{X}$ are lower bounded by a $\delta$ that is allowed to be arbitrarily small, then the smallest singular value is at least $Cm^{d/2} (m\delta)^{\lambda-1}$, where $\lambda$ is the maximum number of elements in $\mathcal{X}$ contained within any ball or cube of an explicitly given radius. This estimate communicates a localization effect of the Fourier transform. While it is sharp, the smallest singular value behaves better than expected for many $\mathcal{X}$, including when we dilate a generic set by parameter $\delta$. We next show that if there is a $\eta$ such that, for each $x\in\mathcal{X}$, the set $\mathcal{X}\setminus\{x\}$ locally consists of at most $r$ hyperplanes whose distances to $x$ are at least $\eta$, then the smallest singular value is at least $C m^{d/2} (m\eta)^r$. For dilations of a generic set by $\delta$, the lower bound becomes $C m^{d/2} (m\delta)^{\lceil (\lambda-1)/d\rceil }$. The appearance of a $1/d$ factor in the exponent indicates that compared to worst case scenarios, the condition number of nonharmonic Fourier transforms is better than expected for typical sets and improve with higher dimensionality.

翻译：考虑一个算子，它对支撑在 $\mathcal{X}\subset[-\frac 12,\frac 12)^d$ 上的离散测度进行傅里叶变换，并将其限制在紧集 $\Omega\subset\mathbb{R}^d$ 上。当 $\Omega$ 是半径为 $m$ 的闭球或边长为 $2m$ 的闭立方体，并且在 $\mathcal{X}$ 满足不同类型的几何假设时，我们给出了该算子最小奇异值的下界。我们首先证明，如果 $\mathcal{X}$ 中点之间的距离以一个可以任意小的 $\delta$ 为下界，则最小奇异值至少为 $Cm^{d/2} (m\delta)^{\lambda-1}$，其中 $\lambda$ 是 $\mathcal{X}$ 中落在任何具有明确给定半径的球或立方体内的最大元素数量。该估计揭示了傅里叶变换的一种局部化效应。虽然该估计是尖锐的，但对于许多 $\mathcal{X}$，包括当我们用参数 $\delta$ 膨胀一个一般集合时，最小奇异值的表现优于预期。我们接着证明，如果存在一个 $\eta$，使得对于每个 $x\in\mathcal{X}$，集合 $\mathcal{X}\setminus\{x\}$ 局部由至多 $r$ 个超平面构成，且这些超平面到 $x$ 的距离至少为 $\eta$，那么最小奇异值至少为 $C m^{d/2} (m\eta)^r$。对于用 $\delta$ 膨胀的一般集合，下界变为 $C m^{d/2} (m\delta)^{\lceil (\lambda-1)/d\rceil }$。指数中出现 $1/d$ 因子表明，与最坏情况相比，非调和傅里叶变换的条件数对于典型集合的表现优于预期，并且随着维度的增加而改善。