Parseval frame wavelets with En(2) dilations

We study Parseval frame wavelets in L-2(R-n) With matrix dilations of the form (Df)(x) = root 2f (Ax), where A is an arbitrary expanding n x it matrix with integer coefficients, such that |det A| = 2. We show that each A-MRA admits either Parseval frame wavelets, or Parseval frame bi-wavelets. The minimal number of generators for a Parseval frame associated with an A-MRA (i.e. I or 2) is determined in terms of a scaling function. All Parseval frame (bi)wavelets associated with A-MRA's are described. We then introduce new classes of filter induced wavelets and bi-wavelets. It is proved that these new classes strictly contain the classes of all A-MRA Parseval frame wavelets and bi-wavelets, respectively. Finally, we demonstrate a method of constructing all filter induced Parseval frame (bi)wavelets from generalized low-pass filters. (c) 2005 Elsevier Inc. All rights reserved.