The Stability Of Some Special Frames

Wavelet analysis is a more and more perfect newly developed theory, which has already been applied to many science research fields. Frames is an important part in research of wavelet analysis and its definition was formally advanced by Duffin and Schaefer when they studied the problems of nonharmonic Fourier expansions in 1952. Frame theory is a central tool in many applied areas. Now it has been widespread used in signal analysis, date compress, sample theory and so on. Stability and approximation were two basic points in research of frames. According the research of the frames stability we could know more about the construction and the character of all different kinds of frames. In this paper we study the stability of several kinds of special frames from different point of views and use different methods . And some useful results are obtained.This paper consists of five chapters.In chapter 1, the author introduce the history of frame development in research of wavelet analysis and the main results in every period .Besides the development guide of the frame research was introduced.In chapter 2, The stability of Banach frames and atomic decomposition. Firstly, the author study the stability of Banach frames from different point of views; then, we study the stability of atomic decomposition inα-modulation Space.In chapter 3, the author discusses the p-frames of Banach space. Two pieces of character were given and its stability been studied. Furthermore, the author discusses the N-frame and M-Riesz basis in Banach space.In chapter 4, the author study the (Ω,μ)-frames and some characters be given.Then, the identities and inequalities of (Ω,μ) -frames and g-frames were discussed and given. Besides, we discuss the stability of g-frames which consists g-Riesz basis.In chapter 5, the author study the approximation of the wavelet frames dual frames by the finite section method and the approximation of the integer oversampled wavelet frames.