Unilateral Shifts in Wavelet Theory and Algorithm

Abstract

The construction of wavelets is usually derived through a multiresolution analysis and formulated as a pair of quadrature mirror filters {h(n), g(n)} In this construction there is a shift structure of subspaces associated with the dilation operator, which is a bilateral shift of infinite multiplicity. We show that associated with each wavelet construction through multiresolution analysis are two unilateral shifts of infinite multiplicity on ι ²(ℤ). The generating wandering subspaces of these shifts, as well as the shifts themselves, are given in terms of the functions h(n) and g(n) The two shifts are duals of each other in a way that allows us to decompose l ²(ℤ) into the direct sum of the two generating wandering subspaces. It is shown that the decomposition of functions via the first of these two unilateral shifts is equivalent to the algorithm used in the wavelet transform. Thus, we provide a Hilbert space setting for the wavelet algorithm, often referred to as the discrete wavelet transform.