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Hybrid Function Systems in the Theory of Uniform Distribution of Sequences

  • Peter Hellekalek
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 23)

Abstract

A hybrid sequence in the multidimensional unit cube is a combination of two or more lower-dimensional sequences of different types. In this paper, we present tools to analyze the uniform distribution of such sequences. In particular, we introduce hybrid function systems, which are classes of functions that are composed of the trigonometric functions, the Walsh functions in base \(\mathbf{b}\), and the \(\mathbf{p}\)-adic functions. The latter are related to the dual group of the p-adic integers, p a prime. We prove the Weyl criterion for hybrid function systems and define a new notion of diaphony, the hybrid diaphony. Our approach generalizes several known concepts and results.

Notes

Acknowledgements

The author would like to thank Markus Neuhauser, NUHAG, University of Vienna, Austria, and RWTH Aachen, Germany, and Harald Niederreiter, University of Salzburg, and RICAM, Austrian Academy of Sciences, Linz, for several helpful comments.

References

  1. 1.
    Dick, J., Pillichshammer, F.: Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge (2010)Google Scholar
  2. 2.
    Grozdanov, V., Nikolova, E., Stoilova, S.: Generalized b-adic diaphony. C. R. Acad. Bulgare Sci. 56(4), 23–30 (2003)Google Scholar
  3. 3.
    Grozdanov, V.S., Stoilova, S.S.: On the theory of b-adic diaphony. C. R. Acad. Bulgare Sci. 54(3), 31–34 (2001)Google Scholar
  4. 4.
    Hellekalek, P.: General discrepancy estimates: the Walsh function system. Acta Arith. 67, 209–218 (1994)Google Scholar
  5. 5.
    Hellekalek, P.: On the assessment of random and quasi-random point sets. In: P. Hellekalek, G. Larcher (eds.) Pseudo and Quasi-Random Point Sets, Lecture Notes in Statistics, vol. 138, pp. 49–108. Springer, New York (1998)Google Scholar
  6. 6.
    Hellekalek, P.: A general discrepancy estimate based on p-adic arithmetics. Acta Arith. 139, 117–129 (2009)Google Scholar
  7. 7.
    Hellekalek, P.: A notion of diaphony based on p-adic arithmetic. Acta Arith. 145, 273–284 (2010)Google Scholar
  8. 8.
    Hellekalek, P., Leeb, H.: Dyadic diaphony. Acta Arith. 80, 187–196 (1997)Google Scholar
  9. 9.
    Hewitt, E., Ross, K.A.: Abstract harmonic analysis. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, second edn. Springer-Verlag, Berlin (1979)Google Scholar
  10. 10.
    Hofer, R., Kritzer, P.: On hybrid sequences built of Niederreiter-Halton sequences and Kronecker sequences. Bull. Austral. Math. Soc. (2011). To appearGoogle Scholar
  11. 11.
    Hofer, R., Larcher, G.: Metrical results on the discrepancy of Halton-Kronecker sequences. Mathematische Zeitschrift (2011). To appearGoogle Scholar
  12. 12.
    Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. John Wiley, New York (1974). Reprint, Dover Publications, Mineola, NY, 2006Google Scholar
  13. 13.
    Mahler, K.: Lectures on diophantine approximations. Part I: g-adic numbers and Roth’s theorem. University of Notre Dame Press, Notre Dame, Ind (1961)Google Scholar
  14. 14.
    Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)Google Scholar
  15. 15.
    Niederreiter, H.: On the discrepancy of some hybrid sequences. Acta Arith. 138(4), 373–398 (2009)Google Scholar
  16. 16.
    Niederreiter, H.: A discrepancy bound for hybrid sequences involving digital explicit inversive pseudorandom numbers. Unif. Distrib. Theory 5(1), 53–63 (2010)Google Scholar
  17. 17.
    Niederreiter, H.: Further discrepancy bounds and an Erdös-Turán-Koksma inequality for hybrid sequences. Monatsh. Math. 161, 193–222 (2010)Google Scholar
  18. 18.
    Schipp, F., Wade, W., Simon, P.: Walsh Series. An Introduction to Dyadic Harmonic Analysis. With the collaboration of J. Pál. Adam Hilger, Bristol and New York (1990)Google Scholar
  19. 19.
    Spanier, J.: Quasi-Monte Carlo methods for particle transport problems. In: H. Niederreiter, P.J.S. Shiue (eds.) Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (Las Vegas, NV, 1994), Lecture Notes in Statist., vol. 106, pp. 121–148. Springer, New York (1995)Google Scholar
  20. 20.
    Zinterhof, P.: Über einige Abschätzungen bei der Approximation von Funktionen mit Gleichverteilungsmethoden. Sitzungsber. Österr. Akad. Wiss. Math.-Natur. Kl. II 185, 121–132 (1976)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SalzburgSalzburgAustria

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