Advertisement

The diaphony and the star-diaphony of some two-dimensional sequences

  • Yi-Jun Xiao
Conference paper

Abstract

In this paper, we estimate the star-diaphony and the diaphony of the Roth sequences, the Zaremba sequences, the Davenport sequences, the symmetric generalized Hammersley sequences and some good lattice points based on their L 2 discrepancy estimates. We show that these sequences have all optimal diaphony.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. Davenport, Note on irregularities of distribution,Mathematika, vol. 3, 131–135 (1956).CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    H. Chaix and H. Faure, Discrépance et diaphonie des suites de van der Corput généralisées,C.R.Acad. Sci. Pans, t. 310, Série I, 315–20 (1990).MathSciNetGoogle Scholar
  3. 3.
    H. Gabai, On the discrepancy of certain sequencesmod 1,Illinois J. Math. 11, 1–12 (1967).MathSciNetGoogle Scholar
  4. 4.
    V. S. Grozdanov, On the diaphony of two-dimensional finite sequences,C.R.Acad, Bulgarae Sci.48, n.4, 15–18 (1995).zbMATHMathSciNetGoogle Scholar
  5. 5.
    J. H. Halton and S. K. Zaremba, The extreme and L2 discrepancies of some plane sets,Monatsh. Math.73, 316–328 (1969).CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    P. Hellekalek, On the assessment of random and quasi-random point sets,Ran- dom and Quasi-Random Point sets, Lecture Notes in Stat., 138, 49–108, 1998.CrossRefMathSciNetGoogle Scholar
  7. 7.
    F. J. Hickernell, A generalized discrepancy and quadrature error bound,Math. Comp.,67, 299–322 (1998).CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    F. J. Hickernell, Lattice rules: how well do they measure up?Random and Quasi- Random Point sets, Lecture Notes in Stat.,138, 109–166, 1998.CrossRefMathSciNetGoogle Scholar
  9. 9.
    J. Hoogland and R. Kleiss, Discrepancy-based error estimates for quasi-Monte Carlo. I: general formalism,Comp. Phys. Comm.,98, 111–127 (1996).CrossRefzbMATHGoogle Scholar
  10. 10.
    L. K. Hua and Y. Wang,Application of number theory to numerical analysis, Springer-Verlag, 1981.Google Scholar
  11. 11.
    L. Kuipers, Simple proof of a theorem of J.F.Koksma,Nieuw Tijdschr. Wisk.55108–111 (1967).Google Scholar
  12. 12.
    V. F. Lev, A diaphony and quadratic discrepancy of multidimensional nets,Math. Remarks, 47, No 6, 45–54 (1990).MathSciNetGoogle Scholar
  13. 13.
    H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers,Bull. AMS. 84, 957–1041 (1978).CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    H. Niederreiter, Existence of good lattice points in the sense of Hlawka,Monatsh. Math. 86, 203–219 (1978).CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    H. Niederreiter,Random number generation and quasi-Monte Carlo methods, SIAM, Philadelphia Pennsylvania, 1992.CrossRefzbMATHGoogle Scholar
  16. 16.
    G. Pagès and Y. J. Xiao, Sequences with low discrepancy and pseudo-random number: theoretical results and numerical tests,J. Statist. Comput. Simul, Vol. 56163–188 (1997).CrossRefzbMATHGoogle Scholar
  17. 17.
    P. D. Proinov, On irregularities of distribution,C.R.Acad, Bulgarae Sci. 39, n. 9, 31–34 (1986).zbMATHMathSciNetGoogle Scholar
  18. 18.
    P. D. Proinov and V. S. Grozdanov, Symmetrization of the van der Corput- Halton sequence,C.R.Acad, Bulgarae Sci. 40, n.8, 5–8 (1987).zbMATHMathSciNetGoogle Scholar
  19. 19.
    K. F. Roth, On irregularities of distribution,Mathematika, vol.1, 73–79 (1954)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    K. F. Roth, On irregularities of distribution, III,Acta. Arithmetica,35, 373–384 (1979).zbMATHMathSciNetGoogle Scholar
  21. 21.
    K. F. Roth, On irregularities of distribution, IV,Acta. Arithmetica,37, 67–75 (1980).zbMATHMathSciNetGoogle Scholar
  22. 22.
    H. Stegbuchner Zur qantitativen Theorie der Gleichverteilung mod 1,Arbeitsberichte Ber. Math. Inst. Univ. Salzburg,3/1980, 9–58 (1980).Google Scholar
  23. 23.
    B. E. White Mean-square discrepancies of the Hammersley and Zaremba sequences for arbitrary radixMonatsh, Math.80, 219–229 1975.CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Y. J. Xiao Suites equireParties associées aux automorPhismes du ToreC.R.Acad. Sci. Paris t.311, SérieI, 579–582 (1990).zbMATHGoogle Scholar
  25. 25.
    S. K. Zaremba L’erreur dans le calcul des inte rales doubles par la methode des bons treillisDemonstratio Math.8347–364 (1975).zbMATHMathSciNetGoogle Scholar
  26. 26.
    P. Zinterhof, Über einige Abschätzungen bei der Aprroximation von Funktionen mit Gleichverteilungsmethoden,S.B. Akad Wiss., math.-natruw. Klasse, Abt. II 185, 121–132 (1976).zbMATHMathSciNetGoogle Scholar
  27. 27.
    P. Zinterhof and H. Stegbuchner, Trigonometrische Approximation mit Gleichverteilungsmethoden,Studia Scientarum Mathematicarum Hungarica,13, 273–289 (1978).zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Yi-Jun Xiao
    • 1
  1. 1.CERMICS-ENPCMarne-la-Vallée Cedex 2France

Personalised recommendations