Frames Of Subspaces In Hilbert Space

Frames for a Hilbert space were formally defined by R.J.Duffin and A.G. Schaeffer in 1952 to study some deep problems in nonharmonic Fourier series. Basically, Duffin and Schaeffer abstracted the fundamental notion of Gabor for studying signal processing.The ideas of Duffin and Schaeffer did not seem to generate much general interest outside of nonharmonic Fourier series however until the landmark paper of Daubechies,Goodmann and Meyer in 1986.After this groundbreaking work,the theory of frames began to be more widely studied.Since the formulation of wavelets theory in 1980s,investigators applied the theory of frames into wavelets analysis to solve the actual problems,which greatly accelerated the development of frame in applications and theories.Traditionally,frames have been used in signal processing,image processing,data compression and sampling theory.Today,ever more uses are being found for the theory such as optics,filterbanks,signal detection,as well as the study of Besov spaces,in Banach space theory etc.In the other direction,powerful tools from operator theory and Banach space theory are being introduced to the study of frames producing deep results in frame theory.At this very moment,the theory is beginning to grow rapidly with the host of new people entering the area.In 2004,general frame theory of subspaces was introduced by Peter G.Casazza ,Gitta Kutyniok and M.Fornasier as a natural generalization of the frame theory in Hilbert spaces to ease the construction of a global frame from local components.In 2005,professor Sun Wenchang studied some special frames (for example bounded quasi-projectors, oblique frames and outer frames),generalized their common properties and gave the definition of g-frames. Frames of subspaces and g-frames are natural generalizations of frames.In this thesis,from the aspect of functiona -1 analysis we will prove they two share many similar properties with abstract frames on the basis of operator theory.This thesis consists of four chapters.In chapter 1: we introduce the development of frames and the popular ways in studying frames,give the basic conceptions of frames,especially we discuss the relationship of Riesz frame and near- Riesz basis. Finally, we introduce the main work of this master thesis.In chapter 2: we mainly study the properties of frames of subspaces.Firstly,we introduce the producing background of frame of subspaces,also concepts such as completeness, minimality,and exactness and basic properties are introduced and investigated.The necessary and sufficient conditions of Riesz decomposition are given,then we prove several basic properties of frame of subspaces.In part 3,we introduce an analysis,a synthesis and a frame operator for a frame of subspaces,which make us improve the proof of proposition 2.5 and extend proposition 2.7 in paper[ll],what important is that we present and prove the perturbations of frame of subspaces by using those operators.Secondly,we discuss the relation between frames of subspaces and the atomic resolution of the identity,then a method of frame constructions is considered,the perturbations of resolution of the indentity are proved.At last, we explore the corresponding relation between the states of preframe operators and the classifications of frames of subspaces.In chapter 3: we derive a fundamental identity for Parseval frames of subspaces. Firstly, we introduce the formulated background of the identity of Parseval frames,then based on operator theory,we generalize the identity to the situation of Parseval frames of subspaces. Several variations of this result are given,including an extension to general frames and the overlapping divisions.Finally,we discuss the derived result,in particular,we derive the equivalent conditions for both sides of the identity to be equal to zero.In chapter 4: we mainly study the related properties of g-frames and a fundamental identity for g-Parseval frames.At first,the definitions of g-frames,frame operator and the dualg-frames are introduced. Moreover ,we extend the notion of dual g-frames and improve the stability of g-frames,then we prove the compact perturbation of g-frames.Secondly,the relation between g-frames and atomic resolution of the identity is given. Finally,we derive a fundamental identity for g-Parseval frames,then we discuss the derived results and get some similar results with above chapter.