L_p- and S_{p,q}^rB-discrepancy of the symmetrized van der Corput sequence and modified Hammersley point sets in arbitrary bases
This work generalizes previous discrepancy estimates for quasi-Monte Carlo methods, offering optimal bounds for a broader class of sequences and point sets.
The authors study the local discrepancy of symmetrized van der Corput sequences and modified Hammersley point sets in arbitrary bases, proving that these achieve the known optimal order for both L_p- and S_{pq}^rB-discrepancy, and provide a sharp upper bound for one-dimensional sequences.
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.