Narrow Results

A Generalized Multiresolution Analysis (GMRA) associated with a wavelet is a sequence of nested subspaces of the function space ℒ²(ℝ), with specific properties, and arranged in such a way that each of the subspaces corresponds to a scale 2^m over all time-shifts n. These subspaces can be expressed in terms of a generating-wandering subspace — of the dyadic-scaling operator — spanned by orthonormal wavelet-functions — generated from the wavelet. In this paper we show that a GMRA can also be expressed in terms of subspaces for each time-shift n over all scales 2^m. This is achieved by means of "elementary" reducing subspaces of the dyadic-scaling operator. Consequently, Time-Shifts GMRA associated with wavelets, as well as "sub-GMRA" associated with "sub-wavelets" will then be introduced.

Let the operators D and T be the dilation-by-2 and translation-by-1 on , which are both bilateral shifts of infinite multiplicity. If ψ(·) in is a wavelet, then {D^mTⁿψ(·)}_{(m,n)∈ℤ²} is an orthonormal basis for the Hilbert space but the reversed set {TⁿD^mψ(·)}_{(n,m)∈ℤ²} is not. In this paper we investigate the role of the reversed functions TⁿD^mψ(·) in wavelet theory. As a consequence, we exhibit an orthogonal decomposition of into T-reducing subspaces upon which part of the bilateral shift T consists of a countably infinite direct sum of bilateral shifts of multiplicity one, which mirrors a well-known decomposition of the bilateral shift D.