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Semi-orthogonal frame wavelets and Parseval frame wavelets associated with GMRA

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  • Liu, Zhanwei
  • Hu, Guoen
  • Wu, Guochang
  • Jiang, Bin

Abstract

In this paper, we study semi-orthogonal frame wavelets and Parseval frame wavelets (PFWs) in L2(Rd) with matrix dilations of form (Df)(x)=2f(Ax), where A is an arbitrary expanding d×d matrix with integer coefficients, such that |detA|=2. Firstly, we obtain a necessary and sufficient condition for a frame wavelet to be a semi-orthogonal frame wavelet. Secondly, we present a necessary condition for the semi-orthogonal frame wavelets. When the frame wavelets are the PFWs, we prove that all PFWs associated with generalized multiresolution analysis (GMRA) are equivalent to a closed subspace W0 for which {Tkψ:k∈Zd} is a Parseval frame (PF). Finally, by showing the relation between principal shift invariant spaces and their bracket function, we discover a property of the PFWs associated with GMRA by the PFWs’ minimal vector-filter. In each section, we construct concrete examples.

Suggested Citation

  • Liu, Zhanwei & Hu, Guoen & Wu, Guochang & Jiang, Bin, 2008. "Semi-orthogonal frame wavelets and Parseval frame wavelets associated with GMRA," Chaos, Solitons & Fractals, Elsevier, vol. 38(5), pages 1449-1456.
    • Handle: RePEc:eee:chsofr:v:38:y:2008:i:5:p:1449-1456
    DOI: 10.1016/j.chaos.2008.04.005
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    Cited by:

    1. Liu, Zhanwei & Hu, Guoen & Lu, Zhibo, 2009. "Parseval frame scaling sets and MSF Parseval frame wavelets," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1966-1974.

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