Perturbation Of Frames, Sampling Theorems And Duality Of Frames

In this thesis, we give some results about the perturbation of frames, Riesz basis and Duality principles of frames in Hilbert spaces or Banach spaces with operator theory in functional analysis. We improve the irregular sampling theorems on the base of improving the perturbations of frames and Riesz basis.This thesis includes five chapers:In chapter 1 we introduce the origin and development of the wavelet analysis and the frame theory.In chapter 2 we study the stability of Riesz frames under perturbation. First, we introduce the basic definitions in the frame theory. Next, we prove the perturbation result of Riesz frames.In chaper 3 we discuss the stability of frames and Riesz basis under perturbation, and these results are applied to the sampling theorems. In the first section we generalize the perturbation of frames and Riesz basis with operator theory in functional analysis. In the second section the previous sampling theorems are improved by the results in the first section.In chapter 4 we discuss the stabilities of frames and Riesz bases under perturbation, and the alternate dual frames of frames. In the first section, we generalize the perturbation of frames and Riesz basis with operator theory in functional analysis. In the second section, using the results in the first section, we improve the previous sampling theorems. In the third section, we prove the equivalent conditions of the alternate dual frames in Hilbert spaces.In chapter 5 we study the duality principles in the frame theory. In the first, second, and third section we generalize the results of Riesz-dual sequences in Hilbert spaces. In the fourth section, we define Riesz-dual sequences in Banach spaces and derive some duality principles of them.