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A wavelet “time-shift-detail” decomposition

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  • Levan, N.
  • Kubrusly, C.S.

Abstract

We show that, with respect to an orthonormal wavelet ψ(·)∈L2(R) any f(·)∈L2(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 L2(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 L2(R) in terms of reducing subspaces of the dilation-by-2 shift operator.

Suggested Citation

  • Levan, N. & Kubrusly, C.S., 2003. "A wavelet “time-shift-detail” decomposition," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 63(2), pages 73-78.
    • Handle: RePEc:eee:matcom:v:63:y:2003:i:2:p:73-78
    DOI: 10.1016/S0378-4754(03)00037-5
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    References listed on IDEAS

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    1. Antoniou, I. & Gustafson, K., 1999. "Wavelets and stochastic processes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 49(1), pages 81-104.
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    1. Kubrusly, Carlos S. & Levan, Nhan, 2004. "Shift reducing subspaces and irreducible-invariant subspaces generated by wandering vectors and applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 65(6), pages 607-627.

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