Some Problems On Hilbert Space R-Dual Theory And Dilation Invariant Systems On The Half Real Line

Motivated by Ron-Shen duality principle and Wexler-Raz biorthogonal rela-tion,Casazza,Kutyniok and Lemmers proposed the concept of R-dual in general Hilbert spaces in 2004.Until now the study on R-dual has been far beyond the research scope in Gabor analysis.It is of independent interest in mathematics,and is an important research area in general frame theory.On the other hand,the study of wavelet and Gabor frames on the real line has seen great achievements during the past more than thirty years,but the study of structured frames on the half real line has been rarely reported.It is because the half real line R₊is not a group under addition,thus the function space L~2(R₊)of square integrable functions on R₊admits no nontrivial wavelet and Gabor system,and L~2(R₊)is not closed under the Fourier transform.This dissertation is divided into two parts:R-dual theory on Hilbert space;a class of dilation invariant systems on the half real line.Chapter 1 is the introductory part that includes concepts,notations,back-grounds and the main results of this dissertation.Chapter 2 focuses on weak R-dual and its duality relations.In this chapter,we introduce the concept of weak R-dual,and investigate the duality relations of weak R-duals.We prove that the weak R-dual of a frame(Riesz basis)is a frame sequence(frame);characterize(unitarily)equivalent frames in terms of weak R-duals;and present an explicit expression of the canonical duals of weak R-duals.Chapter 3 is a continuation of Chapter 2.In this chapter,we introduce the concept of generalized weak R-dual,and prove that the generalized weak R-dual of a frame(Riesz basis)is a frame sequence(frame);canonical duals of general-ized weak R-duals of an arbitrarily given frame does not admit the expression as Chapter 2.And we present a coefficient expression corresponding to the canonical duals of generalized weak R-duals.Chapter 4 focuses on weak g-R-dual and its duality relations.In this chapter,we propose the concept of weak g-R-dual,and investigate its duality relations.We prove that the weak g-R-dual of a g-frame(g-Riesz basis)is a g-frame sequence(g-frame).Using weak g-R-duals we characterize g-frames and(unitary)equivalence between g-frames.And using pseudo-inverse operators we represent the canonical duals of weak g-R-duals.Chapter 5 focuses on HS-R-dual and its duality relations.Since HS-frame is more general than g-frame,we propose the concept of HS-R-dual,and establish the duality relations of HS-R-dual,i.e.,a sequence in B(H,C₂)is an HS-frame(HS-frame sequence,HS-Riesz basis)if and only if its HS-R-dual sequence is an HS-Riesz sequence(HS-frame sequence,HS-Riesz basis).We characterize dual HS-frame pair in terms of HS-R-dual;and prove that,given an HS-frame,among all its dual HS-frames,only the canonical dual admits minimal-norm HS-R-dual.Chapter 6 is a remark on R-duals in Hilbert spaces.Since g-frame is a gener-alization of frame,a natural question to ask is whether g-R-dual is a generalization of R-dual.This chapter gives a negative answer.We introduce the concept of new R-dual(NR-dual),and prove that a g-R-dual is a generalization of new R-dual(not R-dual),and that the NR-dual admits the same duality relations as R-dual.Analogously to Chapter 2,we introduce the concept of NWR-dual,conjecture that NWR-dual and WR-dual need not admit the same duality relations,and reduce the construction of counterexamples to an l~2(N)-operator theoretic problem.Chapter 7 focuses on a class of dilation invariant systems in L~2(R₊).In this chapter,we give a necessary condition for an FGD-system to satisfy the upper frame condition(lower frame condition),and prove that an arbitrary FGD-system cannot be a frame for L~2(R₊).For a class of dilation invariant systems which is a special case of GDI-systems,we present a sufficient condition for them to be Riesz bases(frames)for L~2(R₊).Some examples are also provided.