Abstract
This article provides classes of unitary operators of L2(R) contained in the commutant of the Shift operator, such that for any pair of multiresolution analyses of L2(R) there exists a unitary operator in one of these classes, which maps all the scaling functions of the first multiresolution analysis to scaling functions of the other. We use these unitary operators to provide an interesting class of scaling functions. We show that the Dai-Larson unitary parametrization of orthonormal wavelets is not suitable for the study of scaling functions. These operators give an interesting relation between low-pass filters corresponding to scaling functions, which is implemented by a special class of unitary operators acting on L2([−π, π)), which we characterize. Using this characterization we recapture Daubechies' orthonormal wavelets bypassing the spectral factorization process.
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Communicated by John J. Benedetto
Acknowledgements and Notes. Partially supported by NSF Grant DMS-9157512, and Linear Analysis and Probability Workshop, Texas A&M University Dedicated to the memory of Professor Emeritus Vassilis Metaxas.
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Papadakis, M. Unitary mappings between multiresolution analyses of L2 (R) and a parametrization of low-pass filters. The Journal of Fourier Analysis and Applications 4, 199–214 (1998). https://doi.org/10.1007/BF02475989
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DOI: https://doi.org/10.1007/BF02475989