$L_p$-discrepancy of the symmetrized van der Corput sequence

Abstract

It is well known that the $L_p$-discrepancy for $p\in [1,\infty ]$ of the van der Corput sequence is of exact order of magnitude $O((\log N)/N)$. This however is for $p\in (1,\infty )$ not best possible with respect to the lower bounds according to Roth and Proinov. For the case $p=2$ it is well known that the symmetrization trick due to Davenport leads to the optimal $L_2$-discrepancy rate $O(\sqrt{\log N}/N)$ for the symmetrized van der Corput sequence. In this note we show that this result holds for all $p\in (1,\infty )$. The proof is based on an estimate of the Haar coefficients of the corresponding local discrepancy and on the use of the Littlewood-Paley inequality.

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