Frame Theory And Its Application In Signal Recovery

The information and communication technology is rapidly developing nowadays,and the frames theory provides well support for the development of information and communication technology.Since the establishment of the frames theory,many experts and scholars have systematically studied the properties of frames in Banach and Hilbert spaces,and applied frames theory to solve signal recovery problems,achieving a series of important results.This thesis focuses on the properties of K-frames and improvements to traditional frame-based signal recovery algorithms,which are divided into three main parts:1.The relevant properties of the K-frames were discussed.Firstly,the relevant properties and theorems of frames theory were introduced.The various properties of K-frames under linear mapping,non-negative matrix and specific bounded linear operator were studied.An estimate was made on the frames bound of K-frames,and the relevant properties of dual K-frames under specific perturbation were also studied.2.The stability of K-frames and weaving K-frames and the associated inequalities are studied.Based on the existing perturbation conditions and conclusions,sufficient conditions based on the perturbation conditions of positive bounded series such that the K-frames remains property invariant are investigated and the perturbation is shown to be stable.Sufficient conditions are given for the stability of weaving K-frames in satisfying specific inequality conditions and under the action of n times positive operators,corresponding estimates of the frames bound are given and four corollaries are obtained,and further estimates of the frames bound are discussed for the spectral decomposition of n times positive operators in finite-dimensional Hilbert spaces? on the basis of existing matrix eigenvalue inequalities,the several inequalities under the action of the frames operator and the Gram operator.3.The traditional signal recovery algorithm is optimised.The inverse of the frames operator in the signal recovery process is improved by using the singular value decomposition of the matrix,and numerical experiments show that the improvement improves the transmission efficiency and recovery effect of the signal.The numerical experiments show that the recovery effect is good under the condition of known frames coefficients.