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arXiv · 1507.01684

L_p- and S_{p,q}^rB-discrepancy of the symmetrized van der Corput sequence and modified Hammersley point sets in arbitrary bases

Ralph Kritzinger
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Abstract

We study the local discrepancy of a symmetrized version of the well-known van der Corput sequence and of modified two-dimensional Hammersley point sets in arbitrary base $b$. We give upper bounds on the norm of the local discrepancy in Besov spaces of dominating mixed smoothness $S_{pq}^rB([0,1)^s)$, which will also give us bounds on the $L_p$-discrepancy. Our sequence and point sets will achieve the known optimal order for the $L_p$- and $S_{pq}^rB$-discrepancy. The results in this paper generalize several previous results on $L_p$- and $S_{pq}^rB$-discrepancy estimates and provide a sharp upper bound on the $S_{pq}^rB$-discrepancy of one-dimensional sequences for $r>0$. We will use the $b$-adic Haar function system in the proofs.

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Ralph Kritzinger. 2015-07-07. L_p- and S_{p,q}^rB-discrepancy of the symmetrized van der Corput sequence and modified Hammersley point sets in arbitrary bases. https://arxiv.org/abs/1507.01684

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