The Structure And Properties Of Hidimensional Periodic Wavelets

The construction of wavelets is one of core issues in wavelet analysis. So far, one-dimensional wavelets have seen great achievements, but hidimensional wavelets have not. This is due to the complexity of geometry of the lattices associated with general expansive matrices.In particular, one-dimensional periodic wavelets have interested many waveletters. It is well-known that one-dimensional periodic wavelets can be obtained by periodization of L2(R)-wavelets. A natural problem is whether we can obtain hidimensional periodic wavelets in a similar way.For a general expansive matrix with determinant±2, this thesis focuses on the construction of L2(Td)-wavelets from L2(Rd)-wavelets.Our main results can be stated as follows:Theorem4.1.1. Given a d×d expansive matrix A and φ∈L2(Rd) with φ(0)=1, let φ generate a Parseval frame multiresolution analysis{Vj}j∈Z associated with A. Assume that φ have a bounded radial decreasing L1-majorant R. Then the linear space∞∪j=0Vj is dense in the space L2(Td).Theorem4.2.1. Given a d x d expansive matrix A with|det A|=2, let φ generate a multiresolution analysis{Vj}j∈Z associated with A, and0(0)=1, and let ψ be a corresponding wavelet. Assume that ψ and φ have a common bounded radial decreasing L1-majorant R. Then for j=0,1,2...,(a) Vj (?) Vj+1, Wj (?) Vj+1, and Vj⊕Wj=Vj+1.(b) The system{1, ψj,k:j∈Z+, k∈Γj} is an orthonormal basis of L2(Td).