Multiresolution on compact groups

Abstract

Given a compact group M, we define the notion of multiresolution of L2 (M) with respect to an infinite sequence of subgroups G0 ⊆ G1 ⊆ G2 ⊆ ⋯ such that G = ∪∞ k=0 is a dense subgroup of M. We give characterizations of various axioms of multiresolution, demonstrate the existence and give the construction of a wavelet basis for L2 (M). We also construct stationary multiresolution and wavelets from cyclic vectors. An example of multiresolution on a non-abelian compact group is given for the infinite dihedral group, or isomorphically the real orthogonal group in dimension two. © 1999 Published by Elsevier Science Inc. All rights reserved.