L_p- and S_{p,q}^rB-discrepancy of (order 2) digital nets

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Markhasin, Lev;
  • Identifiers: doi: 10.4064/aa168-2-4
  • Subject: Mathematics - Algebraic Geometry | Mathematics - Numerical Analysis

Dick proved that all order $2$ digital nets satisfy optimal upper bounds of the $L_2$-discrepancy. We give an alternative proof for this fact using Haar bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds of the $S_{p,q}^r B$-discrepancy for ... View more
  • References (25)
    25 references, page 1 of 3

    W. W. L. Chen, M. M. Skriganov, Orthogonality and digit shifts in the classical mean squares problem in irregularities of point distribution. In: Diophantine approximation, 141-159, Dev. Math., 16, Springer, Vienna, 2008.

    H. Davenport, Note on irregularities of distribution. Mathematika 3 (1956), 131-135.

    J. Dick, Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions. SIAM J. Numer. Anal. 45 (2007), 2141-2176.

    J. Dick, F. Pillichshammer, Digital nets and sequences. Discrepancy theory and quasi-Monte Carlo integration. Cambridge University Press, Cambridge, 2010.

    J. Dick, F. Pillichshammer, Optimal L2 discrepancy bounds for higher order digital sequences over the finite field F2. Submitted (2013).

    J. Dick, F. Pillichshammer, Explicit constructions of point sets and sequences with low discrepancy. Submitted (2013).

    H. Faure, F. Pillichshammer, G. Pirsic, W. Ch. Schmid, L2 discrepancy of generalized two-dimensional Hammersley point sets scrambled with arbitrary permutations. Acta Arith. 141 (2010), 395-418.

    G. Halász, On Roth's method in the theory of irregularities of point distributions. Recent progress in analytic number theory, Vol. 2, 79-94. Academic Press, London-New York, 1981.

    A. Hinrichs, Discrepancy of Hammersley points in Besov spaces of dominating mixed smoothness. Math. Nachr. 283 (2010), 478-488.

    A. Hinrichs, Discrepancy, Integration and Tractability. To appear in the Proceedings of MCQMC 2012 (2014).

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