Study Of The Properties And Perturbations Of Some Bases And Frames

In 1952,the concept of frames of Hilbert space was advanced firstly by Duffin R J and Schaeffer A.c, extracting the important ideas in signal processing put forward by Gabor. people have no interest in and payed little attention to frame theory for a long time except for the field of non-harmonic Fourier series.Many researchers applied frame theory to avelet analysis since the subject Wavelet analysis,which is an applying mathematic subject,came into being,particularly when Daubechies etc.found that L2 (R) function could be expanded into a kind of series which is similar to the series expanded by Orthonormal basis in 1986.And gradually the subject of frame was developed in applications and theories. At present, the framework is widely used in many fieldes like Wavelet analysis, signal processing, data compression, samples theory etc. In addition, with the introduction of operator theory and the theory of Banach spaces, frame theory was developed rapidly. Study on the properties and perturbations of bases and framework can make you get a better understanding of the structure and characteristics of them.In this paper we study properties and perturbations of several bases and framework from various angles, applying different methods, and get some meaningful results.This paper consists of five chapters as follows:In chapter 1, the author describes the development process and main search results of the current framework in various stages,besides,the direction of development of the current framework is also introduced.In chapter 2, the author studies frames,Riesz bases, and Riesz frames for Hilbert spaces, some properties and characters are also given.In chapter 3, we introduce the properties and perturbations of g-frame. Furthermore we disscuss the perturbations of g-frame in Hilbert space based on the existing perturbation theory by using operator theory,and promote some existing corresponding results.In chapter 4,using the linear operator theory,we mainly disscuss the perturbations of Banach space and Atomic Decomposition and change the original results in[16,27,41,46]In chapter 5,we introduce the properties and the existing research results of q-frames and p-Riesz basis in Banach space frame and then propose some new perturbation conclusions.