Skip to main content
SpringerLink
Log in
Book cover

Analysis of Divergence pp 51–63Cite as

A substitute for summability in wavelet expansions

  • Chapter

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

Gibbs’phenomenon almost always appears in the expansions using classical orthogonal systems. Various summability methods are used to get rid of unwanted properties of these expansions. Similar problems arise in wavelet expansions, but cannot be solved by the same methods. In a previous work [10], two alternative procedures for wavelet expansions were introduced for dealing with this problem. In this article, we are concerned with the details of the implementation of one of the procedures, which works for the wavelets with compact support in the time domain. Estimates based on this method remove the excess oscillations. We show that the dilation equations which arise, though they contain an infinite number of terms, have coefficients which decrease exponnetially. In addition, an iteration relation for the positive estimation function is derived to reduce the amount of calculation in the approximation. Numerical experiments are given to illustrate the theoretical results.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Bochner and K. Chandrasekharan, Fourier Transforms, Princeton University, Princeton, 1949.

    MATH  Google Scholar 

  2. I. Daubechies, Ten Lectures On Wavelets, CBMS-NSF Series in Appl. Math., SIAM Publ., 1992.

    MATH  Google Scholar 

  3. J. W. Gibbs, Letter To The Editor, Nature (London) 59 (1899), 606.

    Article  Google Scholar 

  4. A.J.E.M. Janssen, The Smith - Barnwell Condition And Non - Negative Scaling Functions, IEEE Trans. Info. The., 38, 884–885 (1994).

    Article  Google Scholar 

  5. S. E. Kelly, Gibbs’ Phenomenon For Wavelets, Appl. Comput. Harmon. Anal. 3(1996), 72–81.

    Article  MathSciNet  MATH  Google Scholar 

  6. S. E. Kelly, M. A. Kon and L. A. Raphael, Local Convergence For Wavelet Expansions, J. Funct. Anal 126(1994), 102–138.

    Article  MathSciNet  MATH  Google Scholar 

  7. Y. Meyer, Ondelettes et Opèrateurs I, Herman, Paris 1990.

    MATH  Google Scholar 

  8. H. T. Shim and H. Volkmer, On Gibbs’ Phenomenon For Wavelet Expansions, J. Approx. The. 84(1996), 74–95.

    Article  MathSciNet  MATH  Google Scholar 

  9. G. G. Walter, Wavelets And Other Orthogonal Wavelets With Applications, CRC Press, 1994.

    MATH  Google Scholar 

  10. G. G. Walter and X. Shen, Positive Estimation With Wavelets, Cont. Math. 216 (1998), 63–79.

    MathSciNet  Google Scholar 

  11. G. G. Walter and H. T. Shim, Gibbs’ Phenomenon For Sampling Series And What To Do About It. J Fourier Anal & Appl, 4 (1998) 355–373.

    MathSciNet  Google Scholar 

  12. S. Zygmund, Trigonometric Series, Cambridge, 1959.

    MATH  Google Scholar 

Download references

Authors
  1. Gilbert G. Walter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoping Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, University of Maine, Orono, ME, 04469, USA

    William O. Bray

  2. Department of Mathematics and Statistics, University of Missouri, Rolla Bldg 202, Rolla, MO, 65409, USA

    Časlav V. Stanojević

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Birkhäuser Boston

About this chapter

Cite this chapter

Walter, G.G., Shen, X. (1999). A substitute for summability in wavelet expansions. In: Bray, W.O., Stanojević, Č.V. (eds) Analysis of Divergence. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2236-1_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-2236-1_5

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-7467-4

  • Online ISBN: 978-1-4612-2236-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation