框架的密度、过完备性和局部化。II。Gabor系统
Density, Overcompleteness, and Localization of Frames. II. Gabor Systems
摘要：
这项工作发展了一个定量的框架来描述一类框架的过完备性。以前的论文介绍了两个框架之间的本地化和近似的概念$\mathcal{F} = \{f_i\}_{i \in I}$和$\mathcal{E} = \{e_j\}_{j \in G}$（$G$离散阿贝尔群），有关的$\mathcal{F}$的元素的扩展通过映射$a \colon I \to G$的$\mathcal{E}$的元素的衰减。本文表明，这些抽象的结果产生了一系列新的影响不规则的Gabor框架。此外，不同的奈奎斯特密度结果的Gabor框架恢复作为特殊情况下，在这个过程中，他们的意义和影响澄清。本文对Gabor框架的超完备性、超完备性、框架边界与密度的关系以及不规则Gabor框架的对偶框架的结构等问题进行了研究，得到了一些新的结果。更一般地说，这些结果适用于伽柏框架和伽柏分子系统，其元素在时间-频率平面中仅共享一个共同的浓度包络。局部化和相关近似属性的概念是一系列概念，它们量化了一个框架的元素可以被另一个框架的元素近似的程度。在本文中，这些本地化和近似概念之间的相互关系进行了全面的检查，大多数的影响是尖锐的。
英文摘要：
This work developes a quantitative framework for describing the overcompleteness of a large class of frames. A previous paper introduced notions of localization and approximation between two frames $\mathcal{F} = \{f_i\}_{i \in I}$ and $\mathcal{E} = \{e_j\}_{j \in G}$ ($G$ a discrete abelian group), relating the decay of the expansion of the elements of $\mathcal{F}$ in terms of the elements of $\mathcal{E}$ via a map $a \colon I \to G$. This paper shows that those abstract results yield an array of new implications for irregular Gabor frames. Additionally, various Nyquist density results for Gabor frames are recovered as special cases, and in the process both their meaning and implications are clarified. New results are obtained on the excess and overcompleteness of Gabor frames, on the relationship between frame bounds and density, and on the structure of the dual frame of an irregular Gabor frame. More generally, these results apply both to Gabor frames and to systems of Gabor molecules, whose elements share only a common envelope of concentration in the time-frequency plane. The notions of localization and related approximation properties are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. In this paper, a comprehensive examination of the interrelations among these localization and approximation concepts is made, with most implications shown to be sharp.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
