$L_p$- and $S_{p,q}^rB$-discrepancy of (order $2$) digital nets
Volume 168 / 2015
Acta Arithmetica 168 (2015), 139-159
MSC: Primary 11K06, 11K38, 42C10, 46E35, 65C05.
DOI: 10.4064/aa168-2-4
Abstract
Dick proved that all dyadic order $2$ digital nets satisfy optimal upper bounds on the $L_p$-discrepancy. We prove this for arbitrary prime base $b$ with an alternative technique using Haar bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds on the discrepancy function in Besov spaces with dominating mixed smoothness for a certain parameter range, and enlarge that range for order $2$ digital nets. The discrepancy function in Triebel–Lizorkin and Sobolev spaces with dominating mixed smoothness is considered as well.
