L-p- and Sp,q(r)B-discrepancy of the symmetrized van der Corput sequence and modified Hammersley point sets in arbitrary bases

摘要

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 (Sp,qB)-B-r([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 (Sp,qB)-B-r-discrepancy. The results in this paper generalize several previous results on L-p- and (Sp,qB)-B-r-discrepancy estimates and provide a sharp upper bound on the (Sp,qB)-B-r-discrepancy of one-dimensional sequences for r > 0. We will use the b-adic Haar function system in the proofs. (C) 2016 Published by Elsevier Inc.