Skip to main content

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1550))

Abstract

The objective of this paper is to introduce and classify various classes of wavelets and to discuss some of their basic properties. In particular, an identity of Littlewood-Paley type is given and is used to partially characterize wavelets and their duals.

Supported by NSF Grant #DMS-89-0-01345 and ARO Contract #DAAL 03-90-G-0091.

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

Access this chapter

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 50.00
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, and B. Ruskai, eds., Wavelets and Their Applications, Jones and Bartlett, Boston (to appear).

    Google Scholar 

  2. C.K. Chui, Introduction to Wavelets, Academic Press, Boston, 1992.

    MATH  Google Scholar 

  3. C.K. Chui, ed., Wavelets: a Tutorial, Academic Press, Boston, 1992.

    MATH  Google Scholar 

  4. C.K. Chui, Wavelets: with Emphasis on Time-Frequency Analysis, Under preparation.

    Google Scholar 

  5. C.K. Chui and C. Li, A general framework of multivariate compactly supported wavelets and dual wavelets, Under preparation.

    Google Scholar 

  6. C.K. Chui and X.L. Shi, Inequalities of Littlewood-Paley type for frames and wavelets, CAT Report #249, Texas A&M University, 1991.

    Google Scholar 

  7. C.K. Chui and X.L. Shi, On a Littlewood-Paley identity and characterization of wavelets, CAT Report #250, Texas A&M University, 1991.

    Google Scholar 

  8. C.K. Chui, J. Stöckler, and J.D. Ward, Compactly supported boxspline wavelets, CAT Report #230, Texas A&M University, 1990.

    Google Scholar 

  9. C.K. Chui and J.Z. Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc, (to appear in 1991).

    Google Scholar 

  10. C.K. Chui and J.Z. Wang, On compactly supported spline wavelets and a duality principle, Trans. Amer. Math. Soc, (to appear in 1991).

    Google Scholar 

  11. C.K. Chui and J.Z. Wang, A general framework of compactly supported splines and wavelets, CAT Report #210, Texas A&M University, 1991.

    Google Scholar 

  12. A. Cohen, Doctoral Thesis, Univ. Paris-Dauphine, 1990.

    Google Scholar 

  13. A. Cohen, I. Daubechies, and J.C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure and Appl. Math. (to appear).

    Google Scholar 

  14. J.M. Combes, A. Grossmann, and Ph. Tchamitchian, eds., Wavelets: Time-Frequency Methods and Phase Space, Springer-Verlag, N.Y., 1989; 2nd Edition, 1991.

    Google Scholar 

  15. I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure and Appl. Math. 41 (1988), pp. 909–996.

    Article  MathSciNet  MATH  Google Scholar 

  16. I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Information Theory 36 (1990), pp. 961–1005.

    Article  MathSciNet  MATH  Google Scholar 

  17. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Math., SIAM Publ., Philadelphia (to appear).

    Google Scholar 

  18. R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), pp. 341–366.

    Article  MathSciNet  MATH  Google Scholar 

  19. J.C. Feauveau, Non-orthogonal multiresolution analysis using wavelets, Wavelets: a Tutorial, ed. by C.K. Chui,, Academic Press, Boston, 1992.

    Google Scholar 

  20. A. Grossmann and J. Morlet, Decomposition of Hardy functions into integrable wavelets of constant shape, SIAM J. Math. Analysis 15 (1984), pp. 723–736.

    Article  MathSciNet  MATH  Google Scholar 

  21. R.Q. Jia and C.A.Micchelli, Using the refinement equation for the construction of pre-wavelets V: extensibility of trigonometric polynomials, Preprint, 1991.

    Google Scholar 

  22. W. Light, ed., Wavelets, Subdivisions, and Radial Functions, Oxford University Press, Oxford, 1991.

    Google Scholar 

  23. R. Lorentz and W. Madych, Wavelets and generalized box splines, Preprint, 1991.

    Google Scholar 

  24. S. Mallat, Multiresolution approximations and wavelets orthonormal bases of L 2 (ℝ), Trans. Amer. Math. Soc. 315 (1989), pp. 69–87.

    MathSciNet  Google Scholar 

  25. S. Mallat and W.L. Hwang, Singularity detection and processing with wavelets, Preprint, 1991.

    Google Scholar 

  26. S. Mallat and S. Zhong, Wavelet transform maxima and multiscale edges, Wavelets and Their Applications (G. Beylkin et. al., Jones and Bartlett, eds.), Boston (to appear).

    Google Scholar 

  27. Y. Meyer, Ondelettes et Opérateurs, (two volumes), Hermann, Paris, 1990.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Andrei A. Gonchar Edward B. Saff

Rights and permissions

Reprints and permissions

Copyright information

© 1993 The Euler International Mathematical Institute

About this paper

Cite this paper

Chui, C.K. (1993). On wavelet analysis. In: Gonchar, A.A., Saff, E.B. (eds) Methods of Approximation Theory in Complex Analysis and Mathematical Physics. Lecture Notes in Mathematics, vol 1550. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0117472

Download citation

  • DOI: https://doi.org/10.1007/BFb0117472

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56931-2

  • Online ISBN: 978-3-540-47792-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics