Advertisement

Gibbs Dyadic Differentiation on Groups - Evolution of the Concept

  • Radomir S. StankovićEmail author
  • Jaakko Astola
  • Claudio Moraga
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10672)

Abstract

Differential operators are usually used to determine the rate of change and the direction of change of a signal modeled by a function in some appropriately selected function space. Gibbs derivatives are introduced as operators permitting differentiation of piecewise constant functions. Being initially intended for applications in Walsh dyadic analysis, they are defined as operators having Walsh functions as eigenfunctions. This feature was used in different generalizations and extensions of the concept firstly defined for functions on finite dyadic groups. In this paper, we provide a brief overview of the evolution of this concept into a particlar class of differential operators for functions on various groups.

References

  1. 1.
    Butzer, P.L., Wagner, H.J.: Approximation by Walsh polynomials and the concept of a derivative. In: Proceedings of Symposium on Applications of Walsh Functions, pp. 388–392, Washington D.C. (1972)Google Scholar
  2. 2.
    Butzer, P.L., Wagner, H.J.: On a Gibbs-type derivative in Walsh-Fourier analysis with applications. In: Proceedings of National Electronic Conference, vol. 27, pp. 393–398 (1972)Google Scholar
  3. 3.
    Butzer, P.L., Wagner, H.J.: Walsh-Fourier series and the concept of a derivative. Appl. Anal. 3, 29–46 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Butzer, P.L., Wagner, H.J.: A calculus for Walsh functions defined on R+. In: Proceedings of Symposium on Applications of Walsh Functions, pp. 75–81, Washington D.C. (1973)Google Scholar
  5. 5.
    Butzer, P.L., Wagner, H.J.: Early contributions from the aachen school to dyadic Walsh analysis with applications to Dyadic PDEs and approximation theory. In: Stankovic, R., et al. (eds.) Dyadic Walsh Analysis from 1924 Onwards Walsh-Gibbs-Butzer Dyadic Differentiation in Science Volume 1 Foundations. ASMES, vol. 12, pp. 161–208. Atlantis Press, Paris (2015).  https://doi.org/10.2991/978-94-6239-160-4_4 CrossRefGoogle Scholar
  6. 6.
    Cooley, J.W., Tukey, J.W.: On the origin and publication of the FFT paper. Curr. Contents ISI Phy. Chem. Earth Sci. 33(51–52), 8–9 (1993)Google Scholar
  7. 7.
    Gibbs, J.E.: Walsh spectrometry, a form of spectral analysis well suited to binary digital computation, p. 24, National Physical Laboratory, Teddington, Middx, UK (1967)Google Scholar
  8. 8.
    Gibbs, J.E., Gebbie, H.A.: Application of Walsh functions to transform spectroscopy. Nature 224(5223), 1012–1013 (1969). publication date 12/1969CrossRefGoogle Scholar
  9. 9.
    Gibbs, J.E., Ireland, B.: Some generalizations of the logical derivative. DES report no. 8, National Physical Laboratory, p. 22+ii, August 1971Google Scholar
  10. 10.
    Gibbs, J.E., Ireland, B.: Walsh functions and differentiation. In: Proceedings of International Conference on Applications of Walsh Functions and Sequency Theory, pp. 147–176 (1974)Google Scholar
  11. 11.
    Gibbs, J.E., Simpson, J.: Differentiation on finite Abelian groups, p. 34, National Physical Laboratory, Teddington, Middx, UK (1974)Google Scholar
  12. 12.
    Haar, A.: Zur theorie der orthogonalen Funktionsysteme. Math. Annal. 69, 331–371 (1910)CrossRefzbMATHGoogle Scholar
  13. 13.
    Onneweer, C.W.: Fractional differentiation on the group of integers of the \(p\)-adic or \(p\)-series field. Anal. Math. 3, 119–130 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Onneweer, C.W.: On the definition of dyadic differentiation. Appl. Anal. 9, 267–278 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pichler, F.R.: Some aspects of a theory of correlation with respect to Walsh harmonic analysis, Technical report R-70-11. University of Maryland Technology Research (1970)Google Scholar
  16. 16.
    Rademacher, H.: Einige Sätze von allgemeinen Orthogonalfunktionen. Math. Ann. 87, 122–138 (1922)CrossRefzbMATHGoogle Scholar
  17. 17.
    Stanković, R.S.: A note on differential operators on finite non-Abelian groups. Appl. Anal. 21(1–2), 31–41 (1986)CrossRefzbMATHGoogle Scholar
  18. 18.
    Stanković, R.S.: Gibbs derivatives on finite non-Abelian groups. In: Butzer, P.L., Stanković, R.S. (eds.), Theory and Applications of Gibbs Derivatives: Proceedings of the First International Workshop on Gibbs Derivatives, 26–28 September 1989, Kupari-Dubrovnik, Yugoslavia, Matematički Institut, Beograd, pp. 269–297 (1990)Google Scholar
  19. 19.
    Stanković, R.S., Astola, J.T.: Gibbs Derivatives - the First Forty Years. TICSP Series #39. Tampere International Center for Signal Processing, Tampere, Finland (2008). ISBN 978-952- 15-1973-4, ISSN 1456-2774Google Scholar
  20. 20.
    Stanković, R.S., Butzer, P.L., Schipp, F., Wade, W.R., Su, W., Endow, Y., Fridli, S., Golubov, B.I., Pichler, F., Onneweer, K.C.W.: Dyadic Walsh Analysis from 1924 Onwards Walsh-Gibbs-Butzer Dyadic Differentiation in Science Volume 1, Foundations. A Monograph Based on Articles of the Founding Authors, Reproduced in Full. Atlantis Studies in Mathematics for Engineering and Science, Vol. 12. Atlantis Press, Springer (2015)Google Scholar
  21. 21.
    Stanković, R.S., Butzer, P.L., Schipp, F., Wade, W.R., Su, W., Endow, Y., Fridli, S., Golubov, B.I., Pichler, F., Onneweer, K.C.W.: Dyadic Walsh Analysis from 1924 Onwards, Walsh-Gibbs-Butzer Dyadic Differentiation in Science, Volume 2, Extensions and Generalizations. A Monograph Based on Articles of the Founding Authors, Reproduced in Full, Atlantis Studies in Mathematics for Engineering and Science. Atlantis Press (2015)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Radomir S. Stanković
    • 1
    Email author
  • Jaakko Astola
    • 2
  • Claudio Moraga
    • 3
  1. 1.Department of Computer ScienceFaculty of Electronic EngineeringNišSerbia
  2. 2.Department of Signal ProcessingTampere University of TechnologyTampereFinland
  3. 3.Faculty of Computer ScienceTechnical University of DortmundDortmundGermany

Personalised recommendations