Perturbation Of Frame And Dilation Property Of A Unitary System

Frames that play an important role in wavelet analysis were firstly defined by R. J. Duffin and A. G. Schaeff'er in 1952 when they studied the non-harmonious Fourier series. Recently D. R. Larson, Deguang Han and Xingde Dai have put forward some definitions such as unitary system, frame vector in terms of applying operator theory to the study of frame theory, which made the research develope rapidly. On this foundation, we introduce the definitions of frame, dilation property of a unitary system and wavelet-type unitary system and study the frame perturbation; the dilation property of some special unitary systems and the basic properties of wavelet-type unitary system. This thesis consists of three chapters:In chapter 1 we discuss the perturbation of frames. In the first section we give two important conclusions of frame perturbation in Hilbert space made by Ole Christensen. According to them, we study the more general case. That's if {fα}α∈A is a frame for a Hilbert space H, then how close to it can make a sequence {gα}α∈A in H be also a frame. In the second section we generalize the definition of frame for a Hilbert space to a Banach space and simplify the proof in [9] in terms of pre-frame operators. That's if {fn}n∈Z is an order q frame for a Banach space X, then how close to it can make a sequence {gn}n∈z in X be also an order q frame. In the third section we firstly introduce the concept of frame vector for a unitary system, then give a if-and-only-if condition for it and at last we get the condition for the sum Aη1 + Bη2 to be a frame vector, where A,B are two operators and η1,η2 are two frame vectors.In chapter 2 we study the dilation property of a unitary system. In the first section we firstly introduce the definition of a Gabor-type unitary system and the dilation property of a unitary system, then prove that some unitary systems have the property such as group unitary system and Gabor-type unitary system. In section 2 we give the definition of a group-like unitary system, define a unitary representation for a unitary system u, ∈ : u → B(12(u)), and give some properties of g = {λu|U ∈ U} : (1) λU/λV = f(UV)λσ(UV);, (2) λU-1 = f(U-1)λσ(U-1). Therefore we know Q is a group-like unitary system. At last we prove that group-like unitary system has the dilation property. In the third section we introduce the definition of wavelet unitary system < D.T >. give a characterization of W(< D.T >) andprove that < D, T > has the dilation property.In chapter 3 we generalize the wavelet unitary system < D.T > to the wavelet type unitary system uD,T-. In the first section we introduce the definitions of uD,T and its frame vector and get some properties as follows: (1) W(UD,T) = {V|V∈ U(C/,(uD,T))}', (2) η is a generalized normal tight wavelet if only and if there exists a co-identity operator A∈C,(uD,T) such that η = A; (3) η is a generalized wavelet frame for uD,T if and 皀ly if there exists a unique operator A ∈ C(uD,T) sucn that η = A and aI ≤ AAη ≤ bI. At last we also prove that uD,T has the dilation property. In the second section we study some basic stucture properties of uD,T: (1) a(uo) is a group in U(C(UD,T)); (2) U(C(UD,T)) is an abelian group.