Study Of The G-Frames And G-Riesz Bases

At present, the frame theory has formed the complete system [2,16], tomultiplex development [43,44,45,48,39,56].For example, Gabor frame [12,30,32],wavelet frame [42,55], Banach frame [46,47land so on. Recently, the generalframe theory has further obtained the promotion and the research in the Hilbertspace, appeared to bounded quasi-projector [4], frame of subspaces [5], outerframe [6],oblique frame[7],pseudo frame[17]and a class of time-frequencylocalization operations [9] and so on. The G- frame proposed in this foundationwith the general significance is the generalized frame, is the general frameconcept extension, it has the very good understanding to iterative solvers ofthe elliptic operator equations, soluting the variational question and thediscretizations of question to having a better application in the processingmathematics.This paper take the general frame theory as a foundation, it make thefurther discussion to the g-frame perturbation, to the g-Riesz base nature, tothe pseudo dual g- frame nature and to approximation of the inverse G-frameoperator and so on.This paper consists of five chaptersIn chapter 1, the author introduces the history of frame development inthe wavelat research process and conditions ,and also the main results in theDomestic and foreign. Then list the main conclusions of this paper.In chapter 2, the author mainly elaborates the g- frame perturbation, thenobtains the perturbation result of the g- frame under tight operator function.In chapter 3, the author mainly discusses the nature of the g-Riesz baseand obtains several better characteristic results. In chapter 4, the author proposes the concept of the pseudo-dual g- framein the general pseudo-dual frame theory foundation, and give the correspondingresearch of its perturbation and its nature.In chapter 5, this part discusses the apperoximation of the Inverse g-frame operator based on the genaral frame apperoximation theory.