A unilateral shift setting for the fast wavelet transform

Wavelets are a basis for L2(R) and the structure of the subspaces involved in a wavelet decomposition of L2(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 l2(Z). The procedure of using the wavelet theory of L2(R) to decompose functions in l2(Z) is commonly referred to as the fast wavelet transform (FWT). In this paper, we describe the structure of subspaces in l2(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 l2(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.