Research On Phase Retrieval In Hilbert Space And Banach Space

Frame is a generalization of orthonormal basis in Hilbert space.It has a wide range of applications in many fields,such as computational science,engineering and so on.In many research topics of frame theory,the phase retrieval of frame is an important and promising frontier problem.In this paper,we study the phase retrieval in Hilbert space and Banach space.In the first chapter,we mainly introduce the background and development of phase retrieval,give the definition of phase retrieval and frame,briefly introduce some properties of the frame,and put forward the problems that need to be studied.In the second chapter,the main research results of phase retrieval in Hilbert space and Banach space are introduced.The three questions are mainly answered,which are the relationship between the phase retrieval and complement properties of frames,whether the phase retrieval frames are stable and whether they are stable under ? disturbance.In the three chapter,we consider the problem of special frames in a separable Hilbert space:let H be a separable Hilbert space and ?={?_n}n?I be a frame of H.is there a ? consisting of ? that is phase recoverable?When ?={e_?:???} is the orthogonal basis of H,is ? still phase recoverable?In view of the above problems,the phase retrieval problem in separable Hilbert space is further studied,and the following conclusions are obtained:let ? be the frame of Hilbert space H,then ? can form a new countable set,? satisfies the phase retrieval and the properties of ? are discussed.In the four chapter,we first introduce the phase retrieval of P-frame in Banach space,and study the phase retrieval problem in real space step by step,and get some useful conclusions.