Semi-orthogonal Frame Wavelets and Parseval Frame Wavelets Associated with GMRA

摘要

In this paper, we study semi-orthogonal frame wavelets and Parseval frame wavelets(PFWs) in L²(R^d) with matrix dilations of form (Df)(x)=2f(Ax), where A is an arbitrary expanding d × d matrix with integer coefficients, such that |detA| = 2. Firstly, we obtain a necessary and sufficient condition for a frame wavelet to be a semi-orthogonal frame wavelet. Secondly, we present a necessary condition for the semi-orthogonal frame wavelets. When the frame wavelets are the PFWs, we prove that all PFWs associated with generalized multiresolution analysis (GMRA) are equivalent to a closed subspace W0 for which {Tk ψ : k ∈ Z^d} is a PFW. Finally, by showing the relation between principal shift invariant spaces and their bracket function, we discover a property of the PFWs associated with GMRA by the PFWs’ minimal vector-filter. In each section, we construct concrete examples.