Theory Of Frames For Banach Spaces

In this article, we introduce the concepts of X_d frames, frames of order p and operator frames, give a series of properties of them and discuss the relations between them and p-frames, Banach frames or X_d—frames. By Banch frames and frames of order p, we establish the theory of reconstruction in Banach spaces. The details as follows:In Chapter 1, we recall these concepts and properties of frames for Hilbert spaces, p-frames, Banach frames and Riesz Bases. By using theory of operators, we characterize some special frames for Hilbert spaces.In Chapter 2, we introduce and study X_d frames, X_d Bessel, tight X_d frames, independent X_d frames and X_d Riesz bases, we also give characterizations of them, necessary and sufficient conditions for Banach space X to have X_d frames or X_d Riesz bases and necessary and sufficient conditions for a X_d frame to have dual frames. With banach frames and frames of order p, which is a special case of X_d frames, we establish wonderful theory of reconstruction (or theory of frame expasion)in Banach spaces.In Chapter 3, we introduce and discuss operator frames, operator Bessel sequences, operator Riesz bases and dual frames of operator frames. We give the char act ilization of operator frames and operator Riesz basis and establish some sufficent and neccessary condition for Banach spaces to have operator frames and operator Riesz basis. We also show that X_d frames and X_d—frames can be regarded as special cases of operators.