Semi-orthogonal frame wavelets and Parseval frame wavelets associated with GMRA

In this paper, we study semi-orthogonal frame wavelets and Parseval frame wavelets (PFWs) in L2(Rd)L2(Rd) with matrix dilations of form (Df)(x)=2f(Ax), where A   is an arbitrary expanding d×dd×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 W0W0 for which {Tkψ:k∈Zd}{Tkψ:k∈Zd} is a Parseval frame (PF). 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.