Journal of Fourier Analysis and Applications

, Volume 6, Issue 6, pp 663–674 | Cite as

Smooth PONS

  • J. S. Byrnes
  • W. Moran
  • B. Saffari


PONStm is a basis which satisfies all of the fundamental properties of the Walsh functions (each element is piecewise constant, takes on only the values ±1, and can be efficiently computed via a fast transform) plus three additional properties that are false for the Walsh functions: PONS is optimal with respect to a global uncertainty principle; all PONS elements have uniformly bounded crest factors; and all PONS elements are QMF’s. In 1991, Ingrid Daubechies asked whether there exists a smooth basis satisfying the global uncertainty principle property. In this article we show how to transform any basis into another basis by applying the PONS construction, thereby providing an affirmative answer to this question.

Math subject classifications

primary 42A65 secondary 41A30 41A58 

Keywords and phrases

PONS Walsh functions crest factor uncertainty principle Shapiro polynomial 


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  1. [1]
    Benedetto, J., Byrnes, J.S., and Shapiro, H.S. (1992). Wavelet auditory models and irregular sampling,Monthly status report, February.Google Scholar
  2. [2]
    Benedetto, J.J. (1990). Uncertainty principle inequalities and spectrum estimation, In Byrnes, J.S. and Byrnes, J.L., Eds.,Recent Advances in Fourier Analysis and its Applications, NATO ASI. Kluwer Academic Publishers, Dordrecht.Google Scholar
  3. [3]
    Benke, G. (1994). Generalized Rudin-Shapiro systems,J. Fourier Anal. and Appl.,1, 87–101.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    Boyd, S. (1986). Multitone signals with low crest factor.IEEE Trans. Cir. & Systems,33, 1018–1022.CrossRefGoogle Scholar
  5. [5]
    Budisin, S.Z., Popovic, B.M., and Indjin, L.M. (1987). Designing radar signals using complementary sequences,Proc. IEE Conf. RADAR 87, 593–597, October.Google Scholar
  6. [6]
    Byrnes, J.S. and Byrnes, J.L. (1989). Recent advances in Fourier analysis and its applications, inProceedings of the NATO Advanced Study Institute, pgs. ix, 79–142, July.Google Scholar
  7. [7]
    Byrnes, J.S. (1994). Quadrature mirror filters, low crest factor arrays, functions achieving optimal uncertainty principle bounds, and complete orthonormal sequences—a unified approach,Applied and Computational Harmonic Analysis, 261–266.Google Scholar
  8. [8]
    Byrnes, J.S., Ramalho, M.A., Ostheimer, G.K., and Gertner, I. (1999). Discrete one dimensional signal processing method and apparatus using energy spreading coding, U.S. Patent number 5,913,186.Google Scholar
  9. [9]
    Byrnes, J.S., Saffari, B., and Shapiro, H.S. (1996). Energy spreading and data compression using the Prometheus orthonormal set, inProc. 1996 IEEE Signal Processing Conf., Loen, Norway.Google Scholar
  10. [10]
    Eliahou, S., Kervaire, M., and Saffari, B. (1991). On Golay polynomial pairs,Advances in Applied Mathematics,12, 235–292.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    Golay, M.J.E. (1949). Multislit spectrometry,J. Optical Society Am.,39, 437.Google Scholar
  12. [12]
    Golay, M.J.E. (1951). Static multislit spectrometry and its application to the panoramic display of infrared spectra,J. Optical Society Am.,41, 468.CrossRefGoogle Scholar
  13. [13]
    Shapiro, H.S. (1958). A power series with small partial sums,Notices of the AMS,6(3), 366.Google Scholar
  14. [14]
    Rudin, W. (1959). Some theorems on Fourier coefficients,Proc. Am. Math. Soc.,10, 855–859.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    Saffari, B. History of Shapiro polynomials and Golay sequences, in preparation.Google Scholar
  16. [16]
    Schroeder, M.R. (1970). Synthesis of low peak factor signals and binary sequences with low autocorrelation,IEEE Trans. Inf. Th.,16, 85–89.CrossRefGoogle Scholar
  17. [17]
    Shapiro, H.S. (1951). Extremal problems for polynomials and power series, Sc.M. thesis, Massachusetts Institute of Technology.Google Scholar

Copyright information

© Birkhäuser 2000

Authors and Affiliations

  • J. S. Byrnes
    • 1
    • 2
  • W. Moran
    • 1
    • 3
    • 4
  • B. Saffari
    • 1
    • 3
    • 4
  1. 1.Prometheus Inc.Newport
  2. 2.University of Massachusetts at BostonBostonUSA
  3. 3.The Flinders University of South AustraliaAustralia
  4. 4.University of ParisOrsay

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