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 $X\subseteq {[-\frac{1}{2},\frac{1}{2})}^d$ and restricts it to a compact $\Omega \subseteq R^d$. We provide lower bounds for its smallest singular value when Ω is either a closed ball of radius m or closed cube of side length 2m, and under different types of geometric assumptions on $X$. We first show that if distances between points in $X$ are lower bounded by a δ that is allowed to be arbitrarily small, then the smallest singular value is at least $C{m}^{d/2}{\left(m\delta \right)}^{\lambda -1}$, where λ is the maximum number of elements in $X$ 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 $X$, including when we dilate a generic set by parameter δ. We next show that if there is a η such that, for each $x\in X$, the set $X\setminus \left\{x\right\}$ locally consists of at most r hyperplanes whose distances to x are at least η, then the smallest singular value is at least $C{m}^{d/2}{\left(m\eta \right)}^r$. For dilations of a generic set by δ, the lower bound becomes $C{m}^{d/2}{\left(m\delta \right)}^{\lceil (\lambda -1)/d\rceil }$. 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.