The Study Of Some Frames For Banach Spaces

Frames theory is the main implement to study wavelet analysis.In1952,frames for Hilbert spaces were introduced by Duffin and Schaeffer as part of their research in nonharmonic Fourier series.In1984,Grossmann found a fundamental new application of frames to wavelet and Gabor transform,which was the starting point for a growing interest in frames.Frames are sequences that have basis-like properties but which need not be bases.In particular,they allow element of a Hilbert space to be written as linear combinations of the frame elements in a stable manner.Because the frame elernents may be linear dependent and therefore the uniqueness of representation characteristic of bases may be lost.This redundancy has important application in,for example,signal and image processing.With the rise and development of wavelet analysis,frames theory gets further improvement.Frames for Banach spaces were brought forward by K.Grochenig in 1991.He generaliszed the concept of frames and defined the general frames for Banach spaces as Banach frames.So frames and atomic decompositions for Banach spaces were developed as a new and important branch of frames theory.This paper is mainly to discuss frames for Banach spaces,and consists of four chapters:Chapter one introduces the rise and development of frames first and the research results of frames theory at home and abroad,then explains the major work done in this paper.Chapter two recalls the basic concepts and properties of Banach frame,X_d -frame p -frames p fram for Hilbert spaces and Banach spaces,and discusss the properties of them further..In chapter three,the author introduces and studies a new frame,X_d frame,studies the conditions for X_d frame equals to operators,and gets several conclusions of stability of X_d frame under the perturbations of operators.At last,the author disusses the properties of X_d Riesz bases.Chapter four disusses the dual frames in Banach spaces first as well as the conditions for the existence of dual frames,we get some meaningful results and then we discuss the independence of frame.