A wavelet "time-shift-detail" decomposition

Authors:
N. Levan;C. S. Kubrusly
Affiliations:
University of California at Los Angeles, Los Angeles, CA;Catholic University of Rio de Janeiro, 22453-900 Rio de Janeiro, RJ, Brazil
Venue:
Mathematics and Computers in Simulation
Year:
2003

Citing 1
Cited 2

Wavelets and stochastic processes

Mathematics and Computers in Simulation

Shift reducing subspaces and irreducible-invariant subspaces generated by wandering vectors and applications

Mathematics and Computers in Simulation
Multiresolution approximation scale and time-shift subspaces

Multidimensional Systems and Signal Processing

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Abstract

We show that, with respect to an orthonormal wavelet ψ(ċ) ∈ L 2 (R) any f (ċ) ∈ L 2 (R) is, on the one hand, the sum of its "layers of details" over all time-shifts, and on the other hand, the sum of its layers of details over all scales. The latter is well known and is a consequence of a wandering subspace decomposition of L 2 (R) which, in turn, resulted from a wavelet multiresolution analysis (MRA). The former has not been discussed before. We show that it is a consequence of a decomposition of L 2 (R) in terms of reducing subspaces of the dilation-by-2 shift operator.