The Translation Function Family And Wavelet System Spanning L~1((?)~d)

Let A be a sequence of Rd and f be a function from Lp(Rd). We say that f is a∧-generator for Lp(Rd) if the system of translates{Tλf|λ∈G∧} spans the space Lp(Rd). We also say that A admits a generator in LP(R). There does exist Z-generators in LP(R) if p> 2. It is well-known that no Z-generators exist in LP(R),1< p< 2. In the space L2(R), it has been shown that on arbitrary perturbation of Z of the form A={n+rn}n∈z, (?)n∈Z,0≠rn, and rn→0, (|n|→∞) admits a generator f∈L2(R). In L1-case, if R(A) = sup{ρ> 0|ε(A) is dense in L2((-ρ,ρ))}=∞, then A admits a generator. The work of this thesis consists of two parts. Part 1 will show which discrete set A admit a generator f in L1(Rd). Part 2 will show whether affine systems can span L1(Rd). The thesis is divided into three chapters.Chapter 1 briefly introduces the background of Fourier analysis, and wavelet analysis.Some auxiliary results and definitions are listed in chapter 2.Chapter 3 presents the main results and the proofs of them.