Study Of Affine Frames And Frames In Shift Invariant Subspaces

Frames for Hilbert space were formally defined by R.J.Duffin and A.G.Sacheffer 1952. Frames for Hilbert space can be viewed as generalized bases for H. Daubechies, Goodmann and Meyer defined affine frame theory in L^p {R)by continual wavelet transformation theory . Charles.K.Chui and Qi Yu Sun gived definition of affine frame operator by operator theory. Functional analysis as an ancient branch of mathmatics is a powerful tool in studing area .In this thesis, we give some results about perturbations of Frames in shift invariant subspaces of L^p (R) and the properties of affine frames.This thesis consists of four chapters.In chapter1: we introduce the development of Frames and the popular ways in studing frames.and the mainly conclusion of this master thesis.In chapter2: we introduce some concepts such as Bessel sequence, Frame, Frame operator .affine Frame and some basic properties of. In part 3 , we give perturbations of q framesin shift subspaces of L^P(R), X(Φ) is defined byif is q frames of X[(Φ), perturbations of {Ψ_j,kk∈Z) will still frames of X(Φ) in some conditions.In chapter3: firstly we introduce some concepts such as affine frame operator, and give mainly conclusion about approximation of canonical dual of Affine Frame operator. We define operator S^-1_n may approximate S^-1, we give some raletive conditions, affine frame operator has the similar property too. In part 2, perturbations of Y(Φ)...