Research On Frame Theory And Its Application In Dealing With Erasures

The frame theory in Hilbert spaces is a branch of wavelet analysis,which is a new research direction combined with operator theory.In essence,frame is a generalization of a basis,it not only inherits the excellent properties of bases,but also has redundancy that bases can not have.It is precisely because of the redundancy of frames,in modern signal transmission systems,frame has gradually replaced the general basis or orthonormal basis as a tool for encoding and decoding.Due to the extensive applications of frames,frame theory has been developed very well since it was proposed.However,there are still many problems which need to be further studied.This thesis is devoted to further research on properties of frames,and focus on signal reconstruction problems by using frames when erasures with probability in the transmission.The main contributions of this dissertation are summarized as follows:1.We investigate some properties of fusion frames in Hilbert spaces.We first give some characterizations of K-fusion frames.We also study the stabilities of K-fusion frames under small perturbations.Then we show that K-fusion frame is equal to atomic system with closed subspace sequence,and construct K-fusion frames from the view of atomic system.Finally we investigate some basic identities with a scalar for fusion frames.These equations are more general,and help to solve the relevant issues of signal reconstruction without phases.2.We investigate some properties of g-frames in Hilbert spaces.Since we do not know that the precise unconditional behavior of g-frame expansions in Hilbert spaces,we first study this problem and use the unconditional(convergence)constant to study the convergence behavior of g-frame expansions.It is shown that the unconditional constants of the g-frame expansion of a vector in a Hilbert space are bounded by a number associated with frame bounds.Then we study the sum and stability of g-frames by using a sequence of positive numbers,and give some conditions for the finite sum of g-frames to be a g-frame.Finally,we introduce a new concept of weaving g-frames,and study the properties of weaving g-frames in Hilbert spaces.We also give some conditions such that a family of g-frames is woven.Two Paley-Wiener type perturbation results for weaving g-frames are given.3.We investigate the problems where frames can be applied to signal recovers where erasures occurred with probability in the transmission process.We first analyze the process of probability loss of coded data,and abstract the transmission of each coded data as a Markov process,and then obtain the loss probabilities of coded data.We construct the Parseval frame by using these probabilities,and call this frame probabilistic modeled Parseval frames.It is shown that probabilistic modeled Parseval frame perform better than traditional Parseval frames for recovering signals with probabilistic erasures.Finally,we present a new decision condition.Based on this condition,we can verify whether a dual frame of a given frame is probability optimal.We also obtain a condition under which the canonical dual frame is probability optimal.Compared with the general optimal dual frame,the probabilistic optimal dual frame can better reduce the effect of probabilistic erasures on the reconstruction result.