Beurling densities and frames of exponentials on the union of small
balls
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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 $\mathbb{R}$.
