框架的冗余度

随着框架理论在信号处理、图像采集、稀疏表示等领域的应用，它越来越受到人们的重视。事实上，使得框架如此有用的正是框架的冗余性，但一直不存在框架冗余度的精确量化定义。直到最近，在无穷维框架里引入了框架测度函数这一工具来研究框架的冗余度，但其不适用于有限维框架的情况。因此，在有限维Hilbert空间里引入了框架冗余函数的定义，并通过上、下冗余度的概念得出了关于框架结构的重要结论。其后，将此冗余函数的概念推广到了无穷维Hilbert空间里。本项目从有限维Hilbert空间框架出发，主要研究以下三个问题：一、进一步研究有限维和无穷维Hilbert空间里框架的冗余度，结构及其在稀疏表示等领域的应用。二、将无穷维情况下得到的结果应用到具体的Gabor框架和小波框架，得到它们的具体结构。三、研究Banach空间里Schauder框架，P-框架等过完备系统的冗余度及结构。

Since the frame theory are very useful in signal processing, image processing and sparse representation, etc, more and more people pay attention to the frame theory. In fact, what makes frames so useful in practice is their redundancy, however, there dose not exist a precise quantitative notion of redundancy. Until recently, somebody introduced the so-called frame measure function in the infinite frames to study the redundancy of frames, but it does not apply to finite frames. Thus，the recent publication introduced the redundancy function, upper redundancy and lower redundancy for finite frames in finite Hilbert space, by these definitions, got the important conclusions about the frames structure.Then,these definitions were generalized to infinite frames. From the finite frames in Hilbert space, the project mainly discusses the following three questions:.1.Further study the redundancy of finite frames and infinite frames, their structures and their applications in sparse representation, etc. .2.The redundancy of Gabor frames and wavelet frames, and their structures..3.The redundancy of overcompetely systems in Banach space, such as Schauder frames,P-frame, etc, and their structures.