Researches On Some Problems Of Frames

Frames of Wavelet Analysis means usually a set consisting of a family of elements that satisfies some properties in a Hilbert space H. Frames has some properties of orthonormal bases, it can be viewed as a " generalized orthonormal bases ". The research of this thesis is on generalized frames in Hilbert spaces, frames in Banach spaces X, and frames in a Hilbert C^*- module H. The research on frames in a Hilbert space comes the following topics: generalized frames for H, frames of subspaces for H, and normalized windowed Fourier transform on L²(R) and L²(Rⁿ). The research on frames in a Banach space space X is on the study of generalized frames for X. Lastly, in light of Hilbert C^*- module, some important properties are also discussed.This article is divided into five chapters: In chapter 1, Bessel sets, normalized tight generalized frames, independent generalized frames, and dual generalized frames are intruduced firstly. Given Bessel set h — {h_m}_m∈M (?) H, using a linear and bounded operator T_h : H → L²(μ), the equivalent properties between T_h, and h are discussed. The stability of generalized frames is studied. Two notions of frame operators and equal-norm tight generalized frames are introduced. The equivalent realtion between generalized frames (resp. normalized generalized frames) and generalized frame operators are discussed. Some properties on equal-norm generalized frames are proved. Finally, some notions of disjointness, strong disjointness of generalized frames, disjoint-preserving operators, and strong disjoint-preserving operators are introduced. Some their important properties are obtained.In chapter 2, we introduce a new concept-Bessel sequence of subspaces, Bessel sequence of subspaces and frames of subspaces for H are investigated. By associating an operator T_W,v: (∑_i∈N(?)W_i)_l²â†’H, it is obtained that a correspondent relationship between Bessel sequence of subspaces (resp. frames of subspaces in H, and operators in B(∑_i∈N(?)W_i)_l²), H). Furthermore, related operator-theoretic characterizations of these sequences are established. As a useful tool, using the norm and inner product to measure " closeness ", stability Theorems of frames of subspaces in H are obtained, i.e, let {W_i}_i∈N is a frame of subspace with respect to {v_i}_i∈N for...