On Krawtchouk Transforms

  • Philip Feinsilver
  • René Schott
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6167)

Abstract

Krawtchouk polynomials appear in a variety of contexts, most notably as orthogonal polynomials and in coding theory via the Krawtchouk transform. We present an operator calculus formulation of the Krawtchouk transform that is suitable for computer implementation. A positivity result for the Krawtchouk transform is shown. Then our approach is compared with the use of the Krawtchouk transform in coding theory where it appears in MacWilliams’ and Delsarte’s theorems on weight enumerators. We conclude with a construction of Krawtchouk polynomials in an arbitrary finite number of variables, orthogonal with respect to the multinomial distribution.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atakishiyev, N.M., Wolf, K.B.: Fractional Fourier-Kravchuk Transform. J. Opt. Soc. Am. A. 14, 1467–1477 (1997)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Atakishiyev, N.M., Pogosyan, G.S., Wolf, K.B.: Finite Models of the Oscillator. Physics of Particles and Nuclei 36(Suppl. 3), 521–555 (2005)Google Scholar
  3. 3.
    Feinsilver, P., Schott, R.: Finite-Dimensional Calculus. Journal of Physics A: Math.Theor. 42, 375214 (2009)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Feinsilver, P., Schott, R.: Algebraic Structures and Operator Calculus. In: Representations and Probability Theory, vol. I-III. Kluwer Academic Publishers, Dordrecht (1993-1995)Google Scholar
  5. 5.
  6. 6.
    Lorente, M.: Orthogonal polynomials, special functions and mathematical physics. Journal of Computational and Applied Mathematics 153, 543–545 (2003)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Lorente, M.: Quantum Mechanics on discrete space and time. In: Ferrero, M., van der Merwe, A. (eds.) New Developments on Fundamental Problems in Quantum Physics, pp. 213–224. Kluwer, Dordrecht (1997) arXiv:quant-ph/0401004v1Google Scholar
  8. 8.
    MacWilliams, F.J., Sloane, N.J.A.: Theory of error-correcting codes. North-Holland, Amsterdam (1977)MATHGoogle Scholar
  9. 9.
    Santhanam, T.S.: Finite-Space Quantum Mechanics and Krawtchuk Functions. In: Proc. of the Workshop on Special Functions and Differential Equations, Madras, India, vol. 192. Allied Publishers, Delhi (1997)Google Scholar
  10. 10.
    Szëgo, Orthogonal Polynomials. AMS, Providence (1955)Google Scholar
  11. 11.
    Yap, P.-T., Paramesran, R.: Image analysis by Krawtchouk moments. IEEE Transactions on Image Processing 12, 1367–1377 (2003)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Philip Feinsilver
    • 1
  • René Schott
    • 2
  1. 1.Southern Illinois UniversityCarbondaleU.S.A.
  2. 2.IECN and LORIANancy-Université, Université Henri PoincaréVandoeuvre-lès-NancyFrance

Personalised recommendations