Revista Matemática Iberoamericana

Size properties of wavelet packets generated using finite filters

Morten Nielsen

Full-text: Open access

Abstract

We show that asymptotic estimates for the growth in $L^p(\mathbb{r})$-norm of a certain subsequence of the basic wavelet packets associated with a finite filter can be obtained in terms of the spectral radius of a subdivision operator associated with the filter. We obtain lower bounds for this growth for $p\gg 2$ using finite dimensional methods. We apply the method to get estimates for the wavelet packets associated with the Daubechies, least asymmetric Daubechies, and Coiflet filters. A consequence of the estimates is that such basis wavelet packets cannot constitute a Schauder basis for $L^p(\mathbb{R})$ for $p\gg 2$. Finally, we show that the same type of results are true for the associated periodic wavelet packets in $L^p[0,1)$.

Article information

Source
Rev. Mat. Iberoamericana, Volume 18, Number 2 (2002), 249-265.

Dates
First available in Project Euclid: 28 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1051544237

Mathematical Reviews number (MathSciNet)
MR1949828

Zentralblatt MATH identifier
1029.42034

Subjects
Primary: 42

Keywords
Wavelet analysis wavelet packets subdivision operators Schauder basis $L^p$-convergence

Citation

Nielsen, Morten. Size properties of wavelet packets generated using finite filters. Rev. Mat. Iberoamericana 18 (2002), no. 2, 249--265. https://projecteuclid.org/euclid.rmi/1051544237


Export citation

References

  • Coifman, R. R., Meyer, Y., Wickerhauser, V.: Size properties of wavelet-packets. In: Wavelets and their applications, 453-470. Jones and Bartlett, 1992.
  • Coifman, R. R., Meyer, Y., Quake, S., Wickerhauser, V.: Signal processing and compression with wavelet packets. In Progress in wavelet analysis and applications, 77-93. Frontières, 1993.
  • Daubechies, I.: Ten lectures on wavelets. Society for Industrial and Applied Mathematics (SIAM), 1992.
  • Goodman, T. N. T., Micchelli, C. A., Ward, J.D.: Spectral radius formulas for subdivision operators. In Recent advances in wavelet analysis, 335-360. Academic Press, 1994.
  • Meyer, Y.: Wavelets and operators. Cambridge University Press, 1992.
  • Séré, É: Localisation fréquentielle des paquets d'ondelettes. Rev. Mat. Iberoamericana 11 (1995), 334-354.
  • Young, R. M.: An introduction to nonharmonic Fourier series. Academic Press Inc., 1980.