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Journal of Fourier Analysis and Applications

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

Smooth PONS

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

Abstract

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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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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