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On convergence of wavelet packet expansions

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Analysis in Theory and Applications

Abstract

It is well known that the-Walsh-Fourier expansion of a function from the block spaceB q([0,1]), 1<q≤∞, converges pointwise a. e. We prove that the same result is true for the expansion of a function fromB q in certain periodized smooth periodic non-stationary wavelet packets bases based on the Haar filters. We also consider wavelet packets based on the Shannon filters and show that the expansion of Lp-functions, 1<p<∞, converges in norm and pointwise almost everywhere.

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References

  1. Daubechies, I., Ten Lectures on Wavelets, Philadelphia, PA: Society for Industrial and Applied Mathematics (SLAM) 1992

    Book  Google Scholar 

  2. Golubov, B., Efimov, A. and Skvortson, V., Walsh Series and Transforms, Dordrecht: Kluwer Academic Publishers Group, Theory and Applications, Translated from the 1987 Russian Original by W. R. Wade, 1991.

    Book  Google Scholar 

  3. Nielsen Hess, N. and Wickerhauser, M. V., Wavelets and Time-Frequency Analsis, Proceedings of the IEEE, 84:4(1996), 523–540.

    Article  Google Scholar 

  4. Meyer, Y., Wavelets and Operators, Cambridge University Press, 1992.

  5. Nielsen, M., Size Properties of Wavelet Packets, Ph.D. Thesis, Washington University, St. Louis, 1999.

    Google Scholar 

  6. Nielsen, M., Walsh-Type Wavelet Packet Expansions, Appl. Comput. Harmon, Anal., 9:3 (2000), 265–285.

    Article  MathSciNet  Google Scholar 

  7. Schipp, F., Wade, W. R. and Simon, P., Walsh Series, Bristol: Adam Hilger Ltd. An Introduction to Dyadic Harmonic Analysis, With the Collaboration of J. Pál, 1990.

  8. Taibleson, N. H. and Weiss, G., Certain Function Spaces Connected With Almost Every-where Convergence of Fourier Series, In: Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. 1, I (Chicago, VI., 1981), Berlmont, CA: Wadworth, 1983, 95–113.

    Google Scholar 

  9. Taibleson, M. H. and Wiess, G., Spaces Generated by Blocks, In: Probability Theory and harmonic Analysis (Cleveland, Ohio 1983), New York: Dekker, 1986, 209–226.

    Google Scholar 

  10. Tao, T., On the Almost Everywhere Convergence of Wavelet Summation Methods, Appl. Comput. Harmon. Anal., 3:4(1996), 384–387.

    Article  MathSciNet  Google Scholar 

  11. Wickerhauser, M. V., Lectures on Wavelet Packet Algorithms, Technical Report, Washington University, 1991.

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Authors and Affiliations

  1. Department of Mathematics, University of South Carolina, 29208, SC, USA

    Morten Nielsen

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  1. Morten Nielsen
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Correspondence to Morten Nielsen.

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Nielsen, M. On convergence of wavelet packet expansions. Approx. Theory & its Appl. 18, 34–50 (2002). https://doi.org/10.1007/BF02837047

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  • DOI: https://doi.org/10.1007/BF02837047

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