L_p- and S_{p,q}^rB-discrepancy of (order 2) digital nets

Abstract

Dick proved that all order $2$ digital nets satisfy optimal upper bounds of the $L_2$-discrepancy. We give an alternative proof for this fact using Haar bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds of the $S_{p,q}^{r}B$-discrepancy for a certain parameter range and enlarge that range for order $2$ digitals nets. $L_p$-, $S_{p,q}^{r}F$- and $S_p^{r}H$-discrepancy is considered as well.

Cited by in corpus (4)

• Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions
• Optimal $L_p$-discrepancy bounds for second order digital sequences
• The curse of dimensionality for the $L_p$-discrepancy with finite $p$
• The $b$-adic symmetrization of digital nets for quasi-Monte Carlo integration