Parseval frame wavelets with [formula omitted]-dilations

Abstract

We study Parseval frame wavelets in L 2 ( R n ) with matrix dilations of the form ( D f ) ( x ) = 2 f ( A x ) , where A is an arbitrary expanding n × n 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. 1 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.