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REVERSED WAVELET FUNCTIONS AND SUBSPACES

NHAN LEVAN

Department of Electrical Engineering, University of California in Los Angeles, Los Angeles, CA 90024-1594, USA

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and

CARLOS S. KUBRUSLY

Catholic University of Rio de Janeiro, 22453-900, Rio de Janeiro, RJ, Brazil

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https://doi.org/10.1142/S0219691307001999Cited by:3 (Source: Crossref)

Abstract

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.

Keywords:

AMSC: 42C40, 47A15

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