Study Of The Banach Frames And Generalied Banach Frames

Frames for a Hilbert space was formally defined by Duffin and Schaefer when they studied the problems of nonharmonic Fourier expansions . Later it became ausefuletool in research of wavelet analysis. Today "frame theory " is a central tool in many applied areas especially in signal analysis where Gabor and Weyl-Heisenberg frames have become a paradigm for the spectral analysis associated with time frequency-methods.The application of operator theory advanced the research of frame theory and a new area of research was found . In this thesis,we use frame theory as a tool to study the frame in a Banach space and Generalied frame in a Banach space, And some useful results are obtained.This thesis consists of five chapters:In the first chapter we formulated the history of frame development in the research of wavelet analysis and the main results in every period. Besides some popular ways in studying frames are introduced in this chapter too.In chapter two, by the use of Pre-frame operator of a Banach frame and its conjugate operator , a sufficient and necessary condition for a Banach frame and dual atomic decomposition is given.In chapter three we study the dual Banach frame and the construction of eachelements in X and X~* under various conditions such as X_d frame, Banach frameand p-frame.In chapter four we study the stability of P -frame and q-Riesz basis.In chapater five , we study the Generalied Banach frame and an equivalent characterization of Generalied Banach frame is given.