Study Of Some Questions For G- Frames

Frames were first introduced in 1952 by Duffin and Schaeffer in the study of nonhar-monic Fourier series. Today, frames not only acquire a great achievement on theoretical research, but also have a rapid development in applications such as image processing, digital communications, etc.. With the deep study of frame theory, some various gener-alizations of frames were given by many authors, particularly, professor Wengchang Sun raised the concept of g-frames in 2006.G-frames are natural generalization of frames which cover all other generalizations of frames. Dual g-frames are important to reconstruction of vectors (or signal) in a Hilbert space. Unfortunately, it is usually difficult to obtain a dual g-frame. However, tight g-frames in a separable Hilbert space have similar properties to that of g-orthonormal bases for expanding arbitrary elements, in addition, approximately dual g-frames and g-dual g-frames could break through the limitations of dual g-frames. So, on the one hand, in this thesis, we study the necessary and sufficient condition for the finite extension to a tight g-frame of a g-frame in an infinite-dimensional Hilbert space and several equivalent conditions for a g-frame in a Hilbert space to be scalable, where the scalable g-frame is a g-frame which can generate a Parseval g-frame by rescaling its frame elements, on the other hand, we introduce the concepts of approximately dual g-frames and g-dual g-frames, find their properties and perturbations. At last, we characterize all g-dual g-frames of a given frame in a Hilbert space. The thesis consists of six chapters.Chapter 1 introduces the background of frames and g-frames and the structure and work of the thesis.Chapter 2 lists the concept and some basic facts of g-frames and the properties of linear bounded operators in a Hilbert space throughout the thesis.Chapter 3 is one of the main contents. The necessary and sufficient conditions for the finite extension to a tight g-frame of a g-Bessel sequence and g-frame in an infinite-dimensional Hilbert space are given.Chapter 4 is the second work. In this chapter several equivalent conditions for a g-frame in a Hilbert space to be scalable are provided.Chapter 5 introduces the concept of approximately dual g-frames and studies their properties and perturbations.The final part is chapter 6. Firstly, the definition of g-dual g-frames is presented, and then the properties and perturbations of g-dual g-frames are given. Finally, the characterization of g-dual g-frames is established.