Study Of Perturbation Of Some Bases And Frames In Banach Spaces

Wavelet analysis is a more and more perfect newly developed theory,which has been applied to many science research fields.Frame is an important part in research of wavelet analysis and its definition was formally advanced by Duffin and Schaeffer when they studied the problems of nonharmonic Fourier series in 1952. Frame which is similar to the nature of the sequence,but not necessarily base.Specifically,they can make the elements of Hilbert space in a stable manner,and the manner is linear.Because the elements of frames may be linear related,they lost the characteristics of expressed uniquely as bases.Also because of this repetitive makes frames have important applications in signal and image processing.So the research of frames has an important practical significance. Stability and approximation are two basic parts of frames research.Through the research on their stability can be a profound understanding on the structures,properties and characteristics of various frames.This paper studies the stability of several frames from different methods and different angles, and obtain some useful results. This paper consists of four chapters as follows:In chapter l,we introduce the history of frame development in research of wavelet analysis and main results in every periods.Besides the research orientations of the frames are also introduced.In chapter 2,we study frames and Riesz frames in Hilbert spaces, some properties and characters are also given.In chapter 3,we introduce the new definitiones of close relations between Banach frames in Banach space and the strong stability about disturbance. Meanwhile we apply the linear bounded operator theory, make further explored about disturbance of Banach frames.In chapter 4, firstly introduce the properties and the existing research results of q-frames and p-Riesz basis in Banach space and then propose some new disturbance conclusions.