论文标题
$ l^2（{\ mathbb r}^{n^2}）$的离散帧$ {\ rm gl} _n（{\ mathbb r}）$
Discrete Frames For $L^2({\mathbb R}^{n^2})$ Arising From Tiling Systems On ${\rm GL}_n({\mathbb R})$
论文作者
论文摘要
$ l^2（{\ Mathbb r}^d）$的离散框架是可计数的序列$ \ {e_j \} _ { \ sum_ {j \ in J} | \ langle f，e_j \ rangle |^2 \ leq b \ | f \ | | _2^2，$ f \ in l^2（\ mathbb {r}^d）$ in l^2 in l^2（\ Mathbb {r}^d）$。我们提出了一种取样连续帧的方法，该方法是由仿射类群的正方形积分表示，以创建用于高维信号的离散帧。我们的方法依赖于使用合适的“瓷砖系统”来分区环境空间。在$ {\ rm m} _n（{\ Mathbb r}）\ rtimes {\ rm gl} _n（{\ mathbb r}）$的情况下，我们提供了构造的所有相关详细信息，尽管此处讨论的方法是一般的，并且可以适应许多其他设置。最后，我们证明，对于$ n = 2 $的情况，我们已经显着改善了框架的界限。
A discrete frame for $L^2({\mathbb R}^d)$ is a countable sequence $\{e_j\}_{j\in J}$ in $L^2({\mathbb R}^d)$ together with real constants $0<A\leq B< \infty$ such that $$ A\|f\|_2^2 \leq \sum_{j\in J}|\langle f,e_j \rangle |^2 \leq B\|f\|_2^2,$$ for all $f\in L^2(\mathbb{R}^d)$. We present a method of sampling continuous frames, which arise from square-integrable representations of affine-type groups, to create discrete frames for high-dimensional signals. Our method relies on partitioning the ambient space by using a suitable "tiling system". We provide all relevant details for constructions in the case of ${\rm M}_n({\mathbb R})\rtimes {\rm GL}_n({\mathbb R})$, although the methods discussed here are general and could be adapted to many other settings. Finally, we prove significantly improved frame bounds over the previously known construction for the case of $n=2$.
