WAVELETS ON COMPACT GROUPS

Abstract

In this project we seek to extend the idea of multiresolution of L2(T). We will give the notion of multiresolution of L2(M); M a compact group, with respect to a MR-group sequence { Gk}?k=0 in M. In the classical case of L2(T) the MR-group sequence is {Z2k}?k=0. We will show how the tools and language in representation theory of finite and compact groups can be used effectively to understand the notion of multiresolution. Characterizations of the three axioms of a multiresolution will be given in Chapter 1. In Chapter 2, we shall look more closely at cyclic vectors of L2( M), introduced in Chapter 1, and describe the structures of these vectors in L2(M). A construction of stationary multiresolution of L2(M) using cyclic vectors will also be given. The decomposition of L2(M) into an orthogonal sum of subspaces via a multiresolution of L2(M) will be discussed in Chapter 3. An explicit method to obtain a wavelet basis for L2( M) will also be given for the case when the multiresolution is stationary. Chapter 4 will give some examples of MR-group sequences and multiresolution; both stationary and non stationary, of L2(T) and L2(D?)