Stability Of Phase Retrievable Continuous Frames For Banach Spaces

Frames have important application in signal and image processing.Phase retrieval of frames is a frontier problem in many framework theory research topics.For discrete frames,phase retrieval of frames is robust in finite dimensional space,while phase retrieval of frames is not robust in infinite dimensional space.Regarding the relationship between frame phase retrieval robustness and frames redundancy,based on the analysis of existing literature,this paper selects a continuous frames suitable for Banach space to discuss the robustness of phase retrievability.In the meantime,we have perfected the relevant conclusions:we give the connection between the strong complement property and the complement property;In addition,by means of the relationship between the lower Lipschitz bound and the ? of strong complement property,we call it the ?-? inequality,and give the relationship between the frame phase recovery robustness and the strong complement property.Finally,it is concluded that the frame phase retrieval robustness and frames redundancy have no relationship,which indicates that increasing the redundancy of the frames can not improve the robustness of the frame phase retrieval.In addition,the discussion about the stability of the phase retrieval frames under ?-disturbance is mostly limited to the frames on Hilbert space.With the introduction of the frames on Banach space,the discussion on the stability of the phase retrieval frames under ?-disturbance is gradually introduced into the frames on Banach space.In this paper,the p~*- frames is selected as the research object.It is found that the p~* -frames with phase recovery are stable under?-disturbance on finite Banach dimensional space,while the p~*- frames with phase retrieval are unstable undere-disturbance on infinite Banach dimensional space.