Study Of The Property Of Some Frames Of Hilbert Space

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 Orthononnal 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 2005, professor Wenchang Sun put forward the concept of the g-frames when study some especial frames(such as external frames and skewed frames),and g-frame is the promotion of general frame. In this paper, we study properties and perturbations of the several bases and framework from various angles, applying different methods, and get some meaningful results.This paper consists of four chapters as follows:In chapter 1,we describe the development process and main search results of the current framework in various stages,and the direction of development of the current framework is also introduced.In chapter 2, we introduce the properties of Bessel frames in Hilbert spacesIn chapter 3, we introduce Riesz bases and Riesz frames in Hilbert spaces, and some properties and perturbations of Riesz frames are also given.In chapter 4, we introduce the conception of g-frames,g-orthonormal basis. Using the linear operator theory, discuss the properties of the lincar operatorS of g-frames in Hilbert space. On this basis, the perturbations of g-frames and stability of g-frames are given, and promote some existing corresponding results.