Research On Frame Theory And Its Application In Data Erasures

As an emerging research direction,frame theory involves many disciplines such as functional analysis,operator theory,signal processing,and coding theory.Similar to the basis,each element in a normed linear space or an inner product space can be represented by a frame.But unlike the basis,the representation is not necessarily unique.Because of this excellent feature of frames,in recent years,the frames are replacing the bases and are widely used in signal processing,transmission channels capacity,coding theory,and other fields.With the development of modern information technology,although the research on frame theory has gradually matured,in order to solve more complex problems,further research on frame theory is necessary.This dissertation is devoted to further exploring the theoretical nature of frames,and the application of frames in data erasures.The specific research contents are divided as follows:(1)The g-frames in Hilbert space are studied.On the one hand,some properties and construction methods of a new type of g-frame,that is,the phase-measurable g-frame,are discussed.Moreover,the perturbation stability and convergence characteristics of phase-recoverable g-frame are studied.We obtain some necessary and sufficient conditions to make a g-frame phase recoverable.On the other hand,we improve the model of the traditional frame operator distance problem and get a new model of the g-frame operator distance problem.The local minimum and global minimum under this model and the relationship between them are discussed respectively.We not only extend part of the properties of the traditional frames to the g-frames but also get the unique properties of the g-frames.These properties help us discover the advantages of g-frames compared to traditional frames so that we can apply them in suitable fields.(2)The K-frames in Hilbert space are studied.On the one hand,we study the sum of several K-frames and its stability from the positive number sequences and the Bessel sequences.The conditions that make the sum of several K-frames still K-frame are obtained.On the other hand,the operator representation of the traditional frame is extended to the K-frame,some properties and stability of the operator representation of the K-frame are explored.Then the conditions that make K-frames and their duals have operator representations are obtained.It is worth noting that,unlike the traditional frames,when the K-frame has an operator representation,the operator used for the representation is inseparably related to the operator K.These properties not only reflect the similarities and differences between the K-frames and traditional frames but also lay the foundation for the application of the K-frames in the following.(3)Research on the application of frame theory in data erasures.Its purpose is to solve the problems in data transmission,aiming at improving the quality of data restoration,and based on the progress made in the previous theory,to provide a better coding system for data transmission.On the one hand,the K-frame is used as the coding frame,and its application in data erasures is studied.Aiming at the problem of probabilistic erasures,we establish a probability erasure model of K-frame,and construct a high-quality coding frame,that is,a probability uniform Parseval K-frame,and prove its existence.Then we get the conditions that make its optimal dual frame the standard K-dual,which can help us simplify the loss problem and reduce the amount of calculation.Next,for the case where the sum of probabilities is not equal to one,the probabilistic loss process is simulated as a Markov process.And in this case,a high-quality coding frame,namely the probability modeled Parseval K-frame is constructed,its existence is proved,and the optimal dual when it is used as an encoding frame is discussed.Then,the reconstruction effects of several types of coding frames are compared with numerical experiments.In fact,the probability modeled Parseval K-frame has more advantages in solving the problem of data probabilistic erasures.On the other hand,the fusion frame is used as the coding frame,and its application in data erasures is studied.Similarly,a probability model for the fusion frame is established and a probability uniform Parseval fusion frame is constructed,and the conditions that make its optimal dual frame the standard dual are obtained.Based on the above discussion,when dealing with the probabilistic erasures,we can choose different encoding and decoding frames according to different situations to make the reconstruction error as small as possible.