Walsh Series Analysis of the L2-Discrepancyof Symmetrisized Point Sets

Abstract. We present a method to estimate the L2-discrepancy of symmetrisized point sets from above and from below with the help of Walsh series analysis. We apply the method to a class of two-dimensional net-type point sets, thereby generalizing results of Halton and Zaremba and of Proinov.

[1]  H. Davenport.  Note on irregularities of distribution , 1956 .

[2]  H. Wozniakowski.  Average case complexity of multivariate integration , 1991 .

[3]  Mean-square discrepancies of the Hammersley and Zaremba sequences for arbitrary radix , 1975 .

[4]  Petko D. Proinov.  Symmetrization of the van der Corput generalized sequences , 1988 .

[5]  M. Skriganov,et al.  Explicit constructions in the classical mean squares problem in irregularities of point distribution , 2002 .

[6]  S. K. Zaremba,et al.  The extreme and L2 discrepancies of some plane sets , 1969 .

[7]  K. F. Roth,et al.  On irregularities of distribution IV , 1979 .

[8]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[9]  W. R. Wade,et al.  Walsh Series, An Introduction to Dyadic Harmonic Analysis , 1990 .

[10]  Peter Hellekalek,et al.  General discrepancy estimates: the Walsh function system , 1994 .

[11]  H. Faure.  Discrépance de suites associées à un système de numération (en dimension s) , 1982 .

[12]  I. Sobol.  On the distribution of points in a cube and the approximate evaluation of integrals , 1967 .

[13]  K. F. Roth.  On irregularities of distribution , 1954 .

[14]  H. Niederreiter.  Point sets and sequences with small discrepancy , 1987 .

[15]  H. Niederreiter,et al.  Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration , 1996 .

[16]  G. Larcher.  Digital Point Sets: Analysis and Application , 1998 .