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Parseval frame wavelets associated with A-FMRA

Author

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  • Wu, Guochang
  • Cheng, Zhengxing
  • Li, Dengfeng
  • Zhang, Fangjuan

Abstract

Mohamed El Naschie’s E-infinity theory has introduced a new framework for understanding and describing nature that is resolution dependent. Wavelets and multiresolution analysis are good mathematical tools to support EI Naschie’s picture of the resolution dependence of the observations. In this paper, inspired by these theory and applications, we study Parseval frame wavelets (PFWs) in L2(Rn) with matrix dilations of the form (Df)(x)=2f(Ax), where A is an arbitrary n×n expanding matrix with integer coefficients, such that ∣detA∣=2. We prove that all PFWs associated to A-FMRA are equivalent to semi-orthogonal Parseval frame wavelets, and characterize all PFWs associated to A-FMRA by showing that they correspond precisely to those for which the dimension function is non-negative integer-valued in L2(Rn). Then, we discover the relation between the spectrum of the central space of an A-FMRA and the supported set of bracket function of its generator and obtain a characterization of PFWs associated with an A-FMRA by the spectrum of the central space of an FMRA. In each section, we construct concrete examples. Thus, we give some mathematical methods to support El Naschie’s picture of the resolution dependence of the observations.

Suggested Citation

  • Wu, Guochang & Cheng, Zhengxing & Li, Dengfeng & Zhang, Fangjuan, 2008. "Parseval frame wavelets associated with A-FMRA," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 1233-1243.
    • Handle: RePEc:eee:chsofr:v:37:y:2008:i:4:p:1233-1243
    DOI: 10.1016/j.chaos.2007.07.075
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    References listed on IDEAS

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    1. El Naschie, M.S., 2006. "Hilbert, Fock and Cantorian spaces in the quantum two-slit gedanken experiment," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 39-42.
    2. Iovane, G., 2006. "Cantorian space–time and Hilbert space: Part II—Relevant consequences," Chaos, Solitons & Fractals, Elsevier, vol. 29(1), pages 1-22.
    3. Iovane, G., 2006. "Cantorian spacetime and Hilbert space: Part I—Foundations," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 857-878.
    4. El Naschie, M.S., 2005. "A guide to the mathematics of E-infinity Cantorian spacetime theory," Chaos, Solitons & Fractals, Elsevier, vol. 25(5), pages 955-964.
    5. Iovane, Gerardo & Giordano, Paola, 2007. "Wavelets and multiresolution analysis: Nature of ε(∞) Cantorian space–time," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 896-910.
    6. El Naschie, M.S., 2006. "Hilbert space, the number of Higgs particles and the quantum two-slit experiment," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 9-13.
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