Beurling densities and frames of exponentials on the union of small balls

TL;DR

This paper characterizes when measures provide frame inequalities for Fourier transforms of functions supported on unions of small balls, linking these conditions to Beurling densities and extending to multiple points and groups.

Contribution

It establishes necessary and sufficient conditions for frame inequalities involving measures and small neighborhoods, introducing a matrix Beurling density for multiple points, and connecting to well-distributed sequences.

Findings

Limits of frame bounds relate to Beurling densities as epsilon approaches zero.

Extension of Beurling density concept to matrix form for multiple points.

Existence of discrete measures satisfying frame inequalities on entire groups.

Abstract

If $x_1,\dots,x_m$ are finitely many points in $\mathbb{R}^d$, let $E_\epsilon=\cup_{i=1}^m\,x_i+Q_\epsilon$, where $Q_\epsilon=\{x\in \mathbb{R}^d,\,\,|x_i|\le \epsilon/2, \, i=1,...,d\}$ and let $\hat f$ denote the Fourier transform of $f$. Given a positive Borel measure $\mu$ on $\mathbb{R}^d$, we provide a necessary and sufficient condition for the frame inequalities $$ A\,\|f\|^2_2\le \int_{\mathbb{R}^d}\,|\hat f(\xi)|^2\,d\mu(\xi)\le B\,\|f\|^2_2,\quad f\in L^2(E_\epsilon), $$ to hold for some $A,B>0$ and for some $\epsilon>0$ sufficiently small. If $m=1$, we show that the limits of the optimal lower and upper frame bounds as $\epsilon\rightarrow 0$ are equal, respectively, to the lower and upper Beurling density of $\mu$. When $m>1$, we extend this result by defining a matrix version of Beurling density. Given a (possibly dense) subgroup $G$ of $\mathbb{R}$, we then consider the problem of characterizing those measures $\mu$ for which the inequalities above hold whenever $x_1,\dots,x_m$ are finitely many points in $G$ (with $\epsilon$ depending on those points, but not $A$ or $B$). We point out an interesting connection between this problem and the notion of well-distributed sequence when $G=a\,\mathbb{Z}$ for some $a>0$. Finally, we show the existence of a discrete set $\Lambda$ such that the measure $\mu=\sum_{\lambda}\,\delta_\lambda$ satisfy the property above for the whole group $\mathbb{R}$.