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

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 $S_{pq}^{r}B([0,1{)}^s)$, 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 $S_{pq}^{r}B$-discrepancy. The results in this paper generalize several previous results on $L_p$- and $S_{pq}^{r}B$-discrepancy estimates and provide a sharp upper bound on the $S_{pq}^{r}B$-discrepancy of one-dimensional sequences for $r>0$. We will use the $b$-adic Haar function system in the proofs.

arXiv comment27 pages, v3: revised version

References in corpus (2)

• Discrepancy of generalized Hammersley type point sets in Besov spaces of dominating mixed smoothness
• Quasi-Monte Carlo methods for integration of functions with dominating mixed smoothness in arbitrary dimension