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

Jean-Pierre Gabardo, Chun-Kit Lai

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 $\hat{f}$ de…