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Hybrid Function Systems in the Theory of Uniform Distribution of Sequences

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Monte Carlo and Quasi-Monte Carlo Methods 2010

Part of the book series: Springer Proceedings in Mathematics & Statistics ((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.

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Salzburg, Hellbrunner Strasse 34, 5020, Salzburg, Austria

    Peter Hellekalek

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  1. Peter Hellekalek
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Correspondence to Peter Hellekalek .

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Editors and Affiliations

  1. , Institute of Applied Mathematics, University of Warsaw, Banacha 2, Warsaw, 02-097, Poland

    Leszek Plaskota

  2. , Institute of Applied Mathematics, University of Warsaw, Banacha 2, Warsaw, 02-097, Poland

    Henryk Woźniakowski

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Hellekalek, P. (2012). Hybrid Function Systems in the Theory of Uniform Distribution of Sequences. In: Plaskota, L., Woźniakowski, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proceedings in Mathematics & Statistics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27440-4_24

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