A unilateral shift setting for the fast wavelet transform
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 a (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. (C) 2000 Published by Elsevier Science B.V. on behalf of IMACS.
unilateral shifts wavelets multiresolution analysis wandering subspaces