框架的密度、过完备性和局部化。I.理论
Density, Overcompleteness, and Localization of Frames. I. Theory
摘要：
这项工作提出了一个定量的框架来描述一个大类的框架的过完备性。它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$\mathcal{F}$的元素的扩展的衰减的$\mathcal{E}$的元素通过映射$a \colon I \to G$。一组基本的等式之间显示三个看似无关的数量：相对措施的$\mathcal{F}$，相对措施的$\mathcal{E}$ -这两个都是由一定的平均值的内积的框架元素与其相应的双框架元素-和密度的一组$a(I)$在$G$。基本的新结果获得的框架的过剩和过完备性，框架之间的关系的界限和密度，并对结构的对偶框架的局部框架。在随后的论文中，这些结果被应用到的情况下的Gabor框架，产生一系列新的结果，以及澄清现有的结果的意义。本文介绍的本地化和相关的近似属性的概念是一个频谱的想法，量化的程度，一个框架的元素可以近似的元素，另一个框架。这些本地化和近似概念之间的相互关系的全面检查。
英文摘要：
This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces 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 $a \colon I \to G$. A fundamental set of equalities are shown between three seemingly unrelated quantities: the relative measure of $\mathcal{F}$, the relative measure of $\mathcal{E}$ - both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements - and the density of the set $a(I)$ in $G$. Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. In a subsequent paper, these results are applied to the case of Gabor frames, producing an array of new results as well as clarifying the meaning of existing results. The notion of localization and related approximation properties introduced in this paper are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. A comprehensive examination of the interrelations among these localization and approximation concepts is presented.
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。
