The Structure Of Shift-invariant Subspaces And The Sampling Theorem

In this thesis, we investigate the structure of shift-invariant subspaces and the sam-pling theorem on them. Let V be a subspace. We say that V is shift-invariant ifwhere the translation operator Tkis defined by （T_kf）（x）=f （x k）. We are interestedin the following three types of shift-invariant subspaces:（i） the shift-invariant subspaces with generators from the super Hilbert spaceL²ï¼ˆR^d）^r,（ii） the shift-invariant subspaces with generators from refinable functions,（iii） the shift-invariant subspaces in Lp（Rd）.In chapter1, we firstly present the background and current status of shift-invariantsubspaces and the sampling theorem. Then the main results of this thesis are stated.In chapter2, we study finitely generated shift-invariant subspaces with generatorsfrom the super Hilbert space L2（Rd）（r）. We give a characterization for the structureof these subspaces. Moreover, we show that every finitely generated shift-invariantsubspace possesses a tight frame. We also give a necessary and sufficient condition forsuch a space to be principal. Our results generalize similar ones for which generatorsare from L2（Rd）.In chapter3, different form many known results where the stability of entries ofrefinable vectors is considered, we study the stability of refinable vectors themselveswhere they are considered as elements of super Hilbert spaces. We call this kind ofstability the vector-stability. We give a necessary and sufficient condition for refinablevectors to be vector-stable. We also give an example to illustrate the difference betweentwo types of stability.In chapter4, we give a characterization of shift-invariant subspaces which are alsoinvariant under additional non-integer translations. Both principal and finitely generat-ed shift-invariant subspaces are studied. Our results improve some known ones. In chapter5, we study the reconstruction of spline functions from their nonuniformsamples. We investigate the existence and uniqueness of the solution of the followingproblem: for given data {（xn,yn）: n∈Z}, find a cardinal spline f （x）, of a given degree,satisfying yn=f （xn）,n∈Z.In chapter6, we discuss the Amrein-Berthier-Benedicks qualitative uncertaintyprinciple and the Donoho-Stark uncertainty principle in the linear canonical transformdomains. Moreover, we derive several necessary or sufficient conditions for discretewindowed linear canonical transform being a frame.