Research On Bessel Vectors, Frame Vectors And Riesz Vectors Of Unitary Systems In A Hilbert Space

Wavelet analysis is a mathematical tool that has been developed and applied in many fields about 10 years recently. It has caused great impact on both conventional sciences and new tecnology subjects.This thesis consists three chapters:Chapter 1: We discuss and study the structure of Bessel vector, frame vector, Riesz vector, introduce the conceptions of B(U),F(U), R(U) and give the characterization of them; obtain properties of local commutant Cψ(U). In part 2 we discuss perturbation of frame vectors and Riesz vectors. In this chapter, the main result is: (1) Uψ∈B(u) C(U) = B(U), (2) Uψ∈F(u){VψSur(Cψ(U))} = F(U), (3) If ψ∈ R(U), then {Vψ : V ∈ Inv(Cψ(U))} = R(U), (4) If [R(U)] = H, and there exists ψ∈ R(U), such that Cψ(U) is an algebra, then for every η∈(U), Cη(U) is Von Neumann algebra.Chapter 2: We introduce the conceptions of TD2π(R.) and DD2(R.), obtain three equivalent conditions for E ∈ TD2π(R)(E ∈ DD2(R)). the sufficient and necessary conditions for E τ～ F(E δ～ F). In part 2 we prove that E is frame wavelet set for a reducedsubspace XΩ if and only if E is basic set and Ω = ∪2nE; E is tiget frame wavelet set for　　　　　　　　　　　　　　　　　　　　　　　　　　n∈Za reduced subspace XΩ if and only if there exists k ≥ 1 such that E = τ(E, 1) = δ(E. k)and Ω =∪ 2nE. In part 3 we get six equivalent conditions for E to be d-wavelet set . n∈ZChapter 3: We introduce the conceptions of translation-Bessel set and translation-wavelet set, characterize them and get that E R is translation-Bessel set for L2(R) ifand only if there exists M > 0 such that μ{ξ∈R : CE(ξ) > M} =0; E R is a translation-wavelet set for L2(R) if and only if there exists a measure 0 set E0 R suchthat R\E0 =∪ tn(E)\E0. tm(E) tn(E) E0 (m ≠ n). n∈Z...