Abstract
Let the operators D and T be the dilation-by-2 and translation-by-1 on [Formula: see text], which are both bilateral shifts of infinite multiplicity. If ψ(·) in [Formula: see text] is a wavelet, then {DmTnψ(·)}(m,n)∈ℤ2is an orthonormal basis for the Hilbert space [Formula: see text] but the reversed set {TnDmψ(·)}(n,m)∈ℤ2is not. In this paper we investigate the role of the reversed functions TnDmψ(·) in wavelet theory. As a consequence, we exhibit an orthogonal decomposition of [Formula: see text] 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.
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