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

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 $$A\Vert f{\Vert }_2^2\le {\int }_{{\mathbb{R}}^d}|\hat{f}(\xi ){|}^{2}d\mu (\xi )\le B\Vert f{\Vert }_2^2,f\in L^2(E_{\epsilon }),$$ 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 $\epsilon \to 0$ 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 $\mathbb{R}$.