A unilateral shift setting for the fast wavelet transform

Abstract

Wavelets are a basis for L 2 ( R) and the structure of the subspaces involved in a wavelet decomposition of L 2 ( R) are well understood and elegantly described by the notion of multiresolution analysis. In practice, however, one is usually more interested in a decomposition of functions in l 2 ( Z) . The procedure of using the wavelet theory of L 2 ( R) to decompose functions in l 2 ( Z) is commonly referred to as the fast wavelet transform (FWT). In this paper, we describe the structure of subspaces in l 2 ( Z) that describes the FWT. We show that for every wavelet constructed through a multiresolution analysis, there corresponds a unilateral shift of infinite multiplicity in l 2 ( Z) such that the decomposition of functions via this unilateral shift is precisely the decomposition of the FWT. In other words, we provide a Hilbert space structure via a unilateral shift to describe the FWT.