Skip to main content
SpringerLink
Log in

Gibbs Phenomenon in Wavelet Analysis

  • Research article
  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

We prove Dirichlet-type pointwise convergence theorems for the wavelet transforms and series of discontinuous functions and we examine the Gibbs ripples close to the jump location. Examples are given of wavelets without ripples, and an example (the Mexican hat) shows that the Gibbs ripple in continuous wavelet analysis can be 3.54% instead of 8.9% of the Fourier case. For the discrete case we show that there exist two Meyer type wavelets the first one has maximum ripple 3.58% and the second 9.8%. Moreover we describe several examples and methods for estimating Gibbs ripples both in continuous and discrete cases. Finally we discuss how a wavelet transform generates a summability method for the Fourier case.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chui, C.K., An Introduction to Wavelets. Academic Press, New York, 1991.

    Google Scholar 

  2. Daubechies, I, Ten Lectures on Wavelets. Capital City press, Vermont, 1992.

    Book  Google Scholar 

  3. Holshneider, M., Wavelets an Analysis Tool. Oxford Science Publication, Oxford, 1995.

    Google Scholar 

  4. Jerri, A.J., Error Analysis in Applications of Generalization of the Sampling Theorem. In: Advanced Topics in Shannon Sampling, R.J. Marks II (ed), Springer, New York, 1993.

    Google Scholar 

  5. Jerri, A.J., Integral and Discrete Transforms With Applications and Error Analysis. Marcel Decker, New York, 1992.

    Google Scholar 

  6. Jerri, A.J., The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations. Kluwer Academic Publishers, (1997 Preprint).

  7. Papoulis, A., The Fourier Integrals and its Applications. McGraw Hill, New York, 1962.

    Google Scholar 

  8. Tichmarsh, E.C., Introduction to the Theory of Fourier Integrals. Oxford University Press, Oxford, 1948.

    Google Scholar 

  9. Walter, G. G., Wavelets and other orthogonal systems with applications. CRC, 1994.

  10. Zygmund, A., Trigonometric Series I and II. 2nd edition, Cambridge Univ. Press, Cambridge, 1959.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics Aristotle, University of Thessaloniki Thessaloniki, 54006, Greece

    C. Karanikas

Authors
  1. C. Karanikas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Karanikas.

Additional information

Dedicated to Professor Paul Butzer for the 70th anniversary of his birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Karanikas, C. Gibbs Phenomenon in Wavelet Analysis. Results. Math. 34, 330–341 (1998). https://doi.org/10.1007/BF03322059

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03322059

Mathematical Subject Classification

Key words

Search

Navigation