IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v52y2000i5p361-379.html
   My bibliography  Save this article

A unilateral shift setting for the fast wavelet transform

Author

Listed:
  • Pham, Joseph N.Q.

Abstract

Wavelets are a basis for L2(R) and the structure of the subspaces involved in a wavelet decomposition of L2(R) are well understood and elegantly described by the notion of multiresolution analysis. In practice, however, one is usually more interested in a decomposition of functions in l2(Z). The procedure of using the wavelet theory of L2(R) to decompose functions in l2(Z) is commonly referred to as the fast wavelet transform (FWT). In this paper, we describe the structure of subspaces in l2(Z) that describes the FWT. We show that for every wavelet constructed through a multiresolution analysis, there corresponds a unilateral shift of infinite multiplicity in l2(Z) such that the decomposition of functions via this unilateral shift is precisely the decomposition of the FWT. In other words, we provide a Hilbert space structure via a unilateral shift to describe the FWT.

Suggested Citation

  • Pham, Joseph N.Q., 2000. "A unilateral shift setting for the fast wavelet transform," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 52(5), pages 361-379.
  • Handle: RePEc:eee:matcom:v:52:y:2000:i:5:p:361-379
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475400001592
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    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. Antoniou, I. & Gustafson, K., 1999. "Wavelets and stochastic processes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 49(1), pages 81-104.
    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. Barunik, Jozef & Krehlik, Tomas & Vacha, Lukas, 2016. "Modeling and forecasting exchange rate volatility in time-frequency domain," European Journal of Operational Research, Elsevier, vol. 251(1), pages 329-340.
    2. Miloš Milovanović & Nicoletta Saulig, 2022. "An Intensional Probability Theory: Investigating the Link between Classical and Quantum Probabilities," Mathematics, MDPI, vol. 10(22), pages 1-16, November.
    3. Miloš Milovanović, 2023. "The Measurement Problem in Statistical Signal Processing," Mathematics, MDPI, vol. 11(22), pages 1-13, November.
    4. Miloš Milovanović & Nicoletta Saulig, 2024. "The Duality of Psychological and Intrinsic Time in Artworks," Mathematics, MDPI, vol. 12(12), pages 1-14, June.
    5. Gialampoukidis, I. & Gustafson, K. & Antoniou, I., 2014. "Time operator of Markov chains and mixing times. Applications to financial data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 415(C), pages 141-155.
    6. 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.
    7. Barunik, Jozef & Vacha, Lukas, 2018. "Do co-jumps impact correlations in currency markets?," Journal of Financial Markets, Elsevier, vol. 37(C), pages 97-119.
    8. Jozef Barunik & Lukas Vacha, 2015. "Realized wavelet-based estimation of integrated variance and jumps in the presence of noise," Quantitative Finance, Taylor & Francis Journals, vol. 15(8), pages 1347-1364, August.
    9. Gialampoukidis, I. & Gustafson, K. & Antoniou, I., 2013. "Financial Time Operator for random walk markets," Chaos, Solitons & Fractals, Elsevier, vol. 57(C), pages 62-72.
    10. Miloš Milovanović & Srđan Vukmirović & Nicoletta Saulig, 2021. "Stochastic Analysis of the Time Continuum," Mathematics, MDPI, vol. 9(12), pages 1-20, June.
    11. Gialampoukidis, Ilias & Antoniou, Ioannis, 2015. "Age, Innovations and Time Operator of Networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 432(C), pages 140-155.
    12. 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.

    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:matcom:v:52:y:2000:i:5:p:361-379. 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: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    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.