Abstract
Krawtchouk matrices have as entries values of the Krawtchouk polynomials for nonnegative integer arguments. We show how they arise as condensed Sylvester-Hadamard matrices via a binary shuffling function. The underlying symmetric tensor algebra is then presented.
To advertise the breadth and depth of the field of Krawtchouk polynomials / matrices through connections with various parts of mathematics, some topics that are being developed into a Krawtchouk Encyclopedia are listed in the concluding section. Interested folks are encouraged to visit the website http://chanoir.math.siu.edu/Kravchuk/index.html which is currently in a state of development.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
N.M. Atakishiyev and K.B. Wolf, Fractional Fourier-Kravchuk transform J. Opt. Soc. Amer. A,147 (1997) 1467–1477.
N. Bose, Digital filters: theory and applications, North-Holland, 1985.
N. Botros, J. Yang, P. Feinsilver, and R. Schott, Hardware Realization of Krawtchouk Transform using VHDL Modeling and FPGAs, IEEE Transactions on Industrial Electronics, 496 (2002)1306–1312.
W.Y.C. Chen and J.D. Louck, The combinatorics of a class of representation functions, Adv. in Math., 140 (1998) 207–236.
P. Delsarte, Bounds for restricted codes, by linear programming, Philips Res. Reports, 27 (1972) 272–289.
P. Delsarte, Four fundamental parameters of a code and their combinatorial significance, Info. & Control, 23 (1973) 407–438.
P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Research Reports Supplements, No. 10, 1973.
C.F. Dunkl, A Krawtchouk polynomial addition theorem and wreath products of symmetric groups, Indiana Univ. Math. J., 25 (1976) 335–358.
C.F. Dunkl and D.F. Ramirez, Krawtchouk polynomials and the symmetrization of hypergraphs, SIAM J. Math. Anal., 5 (1974) 351–366.
G.K. Eagelson, A characterization theorem for positive definite sequences of the Krawtchouk polynomials, Australian J. Stat, 11 (1969) 29–38.
P. Feinsilver and R. Fitzgerald, The spectrum of symmetric Krawtchouk matrices, Lin. Alg. & Appl., 235 (1996) 121–139.
P. Feinsilver and J. Kocik, Krawtchouk matrices from classical and quantum random walks, Contemporary Mathematics, 287 (2001) 83–96.
P. Feinsilver and R. Schott, Krawtchouk polynomials and finite probability theory, Probability Measures on Groups X, Plenum (1991) 129–135.
F.G. Hess, Alternative solution to the Ehrenfest problem, Amer. Math. Monthly, 61 (1954) 323–328.
M. Kac, Random Walks and the theory of Brownian motion, Amer. Math. Monthly 54, 369–391, 1947.
S. Karlin, J. McGregor, Ehrenfest Urn Model, J. Appl. Prob. 2, 352–376, 1965.
M. Krawtchouk, Sur une generalisation des polynomes d’Hermite, Comptes Rendus, 189 (1929) 620–622.
M. Krawtchouk, Sur la distribution des racines des polynomes orthogonaux, Comptes Rendus, 196 (1933) 739–741.
V.I. Levenstein, Krawtchouk polynomials and universal bounds for codes and design in Hamming spaces, IEEE Transactions on Information Theory, 415 (1995) 1303–1321.
S.J. Lomonaco, Jr., A Rosetta Stone for quantum mechanics with an introduction to quantum computation, http://www.arXiv.org/abs/quant-ph/0007045
F.J. MacWilliams and N.J.A. Sloane, The theory of Error-Correcting Codes, The Netherlands, North Holland, 1977.
A.J.F. Siegert, On the approach to statistical equilibrium, Phys. Rev., 76 (1949), 1708–1714.
D. Stanton, Some q-Krawtchouk polynomials on Chevalley groups, Amer. J. Math., 102 (1980) 625–662.
G. Szegö, Orthogonal Polynomials, Colloquium Publications, Vol. 23, New York, AMS, revised eddition 1959, 35–37.
D. Vere-Jones, Finite bivariate distributions and semi-groups of nonnegative matrices, Q. J. Math. Oxford, 222 (1971) 247–270.
M. Voit, Asymptotic distributions for the Ehrenfest urn and related random walks, J. Appl. Probab., 33 (1996) 340–356.
R.K. Rao Yarlagadda and J.E. Hershey, Hadamard matrix analysis and synthesis: with applications to communications and signal/image processing, Kluwer Academic Publishers, 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer Science + Business Media, Inc.
About this chapter
Cite this chapter
Feinsilver, P., Kocik, J. (2005). Krawtchouk Polynomials and Krawtchouk Matrices. In: Baeza-Yates, R., Glaz, J., Gzyl, H., Hüsler, J., Palacios, J.L. (eds) Recent Advances in Applied Probability. Springer, Boston, MA. https://doi.org/10.1007/0-387-23394-6_5
Download citation
DOI: https://doi.org/10.1007/0-387-23394-6_5
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-23378-9
Online ISBN: 978-0-387-23394-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)