论文标题
在可正态组中的近似晶格上的相干系统
Coherent systems over approximate lattices in amenable groups
论文作者
论文摘要
让$ g $为第二个可容纳的组，具有均匀的$ k $ - Approximate lattice $λ$。对于投影离散的系列表示形式$（π，\ MATHCAL {h}_π）正式度$d_π> 0 $的$ g $，我们表明$ d^ - （λ）\ geqd_π / k $对于相干系统$π（λ）g $是$ \ mathcal calcal的完整的，此外，我们表明，如果$π（λ^2）g $很少，则$ d^+（λ^2）\ leqd_πk$。两种必要条件都恢复了均匀晶格的尖锐密度定理，即使对于$ l^2（\ Mathbb {r}）$的Gabor系统也是新的。作为方法的应用，我们还获得了与一般离散集相关的相干帧和riesz序列的必要密度条件。所有结果对于可能的指数增长的可及的单型群有效。
Let $G$ be a second-countable amenable group with a uniform $k$-approximate lattice $Λ$. For a projective discrete series representation $(π, \mathcal{H}_π)$ of $G$ of formal degree $d_π > 0$, we show that $D^-(Λ) \geq d_π / k$ is necessary for the coherent system $π(Λ) g$ to be complete in $\mathcal{H}_π$. In addition, we show that if $π(Λ^2) g$ is minimal, then $D^+ (Λ^2) \leq d_π k$. Both necessary conditions recover sharp density theorems for uniform lattices and are new even for Gabor systems in $L^2 (\mathbb{R})$. As an application of the approach, we also obtain necessary density conditions for coherent frames and Riesz sequences associated to general discrete sets. All results are valid for amenable unimodular groups of possibly exponential growth.
