Nonharmonic multivariate Fourier transforms and matrices: condition numbers and hyperplane geometry

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.