REVERSED WAVELET FUNCTIONS AND SUBSPACES

摘要

Let the operators D and T be the dilation-by-2 and translation-by-1 on L{sup}2(R), which are both bilateral shifts of infinite multiplicity. If ψ(·) in L{sup}2(R) is a wavelet, then {D{sup}m T{sup}nψ(·)}({sub}((m,n)∈Z{sup}2) is an orthonormal basis for the Hilbert space L{sup}2(R) but the reversed set {T{sup}nD{sup}mψ(·)}{sub}((n,m)∈Z{sup}2) is not. In this paper we investigate the role of the reversed functions T{sup}nD{sup}mψ(·) in wavelet theory. As a consequence, we exhibit an orthogonal decomposition of L{sup}2(R) 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.