IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v37y2008i4p1233-1243.html
   My bibliography  Save this article

Parseval frame wavelets associated with A-FMRA

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077907005905
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2007.07.075?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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 spacetime and Hilbert space: Part I—Foundations," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 857-878.
    3. 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.
    4. Iovane, G., 2006. "Cantorian space–time and Hilbert space: Part II—Relevant consequences," Chaos, Solitons & Fractals, Elsevier, vol. 29(1), pages 1-22.
    5. 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.
    6. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Iovane, Gerardo, 2008. "The set of prime numbers: Symmetries and supersymmetries of selection rules and asymptotic behaviours," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 950-961.
    2. Iovane, Gerardo, 2009. "The set of prime numbers: Multiscale analysis and numeric accelerators," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1953-1965.
    3. Sun, Lei & Zhang, Xiaozhong, 2009. "A note on biorthogonality of the scaling functions with arbitrary matrix dilation factor," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 711-715.
    4. Chen, Qing-Jiang & Qu, Xiao-Gang, 2009. "Characteristics of a class of vector-valued non-separable higher-dimensional wavelet packet bases," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1676-1683.
    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. Wang, Xiao-Feng & Gao, Hongwei & Jinshun, Feng, 2009. "The characterization of vector-valued multivariate wavelet packets associated with a dilation matrix," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 1959-1966.
    7. Iovane, Gerardo, 2008. "The distribution of prime numbers: The solution comes from dynamical processes and genetic algorithms," Chaos, Solitons & Fractals, Elsevier, vol. 37(1), pages 23-42.
    8. Sun, Lei & Cheng, Zhengxing & Huang, Yongdong, 2007. "Construction of trivariate biorthogonal compactly supported wavelets," Chaos, Solitons & Fractals, Elsevier, vol. 34(5), pages 1412-1420.
    9. 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.
    10. EL-Nabulsi, Ahmad Rami, 2009. "Fractional action-like variational problems in holonomic, non-holonomic and semi-holonomic constrained and dissipative dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 52-61.
    11. Sergeyev, Yaroslav D., 2007. "Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 50-75.
    12. Iovane, G., 2007. "Hypersingular integral equations, Kähler manifolds and Thurston mirroring effect in ϵ(∞) Cantorian spacetime," Chaos, Solitons & Fractals, Elsevier, vol. 31(5), pages 1041-1053.
    13. Chen, Qingjiang & Huo, Ailian, 2009. "The research of a class of biorthogonal compactly supported vector-valued wavelets," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 951-961.
    14. Sun, Lei & Li, Gang, 2009. "Generalized orthogonal multiwavelet packets," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2420-2424.
    15. Liu, Zhanwei & Hu, Guoen & Wu, Guochang, 2009. "Frame scaling function sets and frame wavelet sets in Rd," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2483-2490.
    16. Chen, Qingjiang & Shi, Zhi, 2008. "Construction and properties of orthogonal matrix-valued wavelets and wavelet packets," Chaos, Solitons & Fractals, Elsevier, vol. 37(1), pages 75-86.
    17. Han, Jincang & Cheng, Zhengxing, 2009. "On the splitting trick and wavelets packets with arbitrary dilation matrix of L2(Rs)," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 130-137.
    18. Iovane, G., 2006. "Cantorian spacetime and Hilbert space: Part I—Foundations," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 857-878.
    19. Wu, Guochang & Li, Zhiqiang & Cheng, Zhengxing, 2009. "Construction of wavelets with composite dilations," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2447-2456.
    20. Li, Rui & Wu, Guochang, 2009. "The orthogonal interpolating balanced multiwavelet with rational coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 892-899.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:37:y:2008:i:4:p:1233-1243. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.