Dick proved that all dyadic order 2 digital nets satisfy optimal upper bounds on the -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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa168-2-4,
author = {Lev Markhasin},
title = {$L\_p$- and $S\_{p,q}^rB$-discrepancy of (order 2) digital nets},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {139-159},
zbl = {1328.11083},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa168-2-4}
}
Lev Markhasin. $L_p$- and $S_{p,q}^rB$-discrepancy of (order 2) digital nets. Acta Arithmetica, Tome 168 (2015) pp. 139-159. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa168-2-4/