Classification of generalized multiresolution analyses

We discuss how generalized multiresolution analyses (GMRAs), both classical and those defined on abstract Hilbert spaces, can be classified by their multiplicity functions m and matrix-valued filter functions H. Given a natural number valued function m and a system of functions encoded in a matrix H satisfying certain conditions, a construction procedure is described that produces an abstract GMRA with multiplicity function m and filter system H. An equivalence relation on GMRAs is defined and described in terms of their associated pairs (m,H). This classification system is applied to MRAs and other classical examples in L2(Rd ) as well as to previously studied abstract examples. © 2009 Elsevier Inc. All rights reserved