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 L-2(R-d) with matrix dilations of form (Df)(x) = root 2f(Ax), where A is an arbitrary expanding d x d matrix with integer coefficients, such that vertical bar detA vertical bar = 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 W-0 for which {T-k psi : k is an element of Z(d)} 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. (C) 2008 Elsevier Ltd. All rights reserved.