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

Lev Markhasin

doi:10.4064/aa168-2-4

Geodesic

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

Lev Markhasin ¹

¹ Institut für Stochastik und Anwendungen Universität Stuttgart Pfaffenwaldring 57 70569 Stuttgart, Germany

Acta Arithmetica, Tome 168 (2015) no. 2, pp. 139-159.

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

Résumé

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.

DOI : 10.4064/aa168-2-4

Keywords: dick proved dyadic order digital nets satisfy optimal upper bounds p discrepancy prove arbitrary prime base alternative technique using haar bases furthermore prove digital nets satisfy optimal upper bounds discrepancy function besov spaces dominating mixed smoothness certain parameter range enlarge range order digital nets discrepancy function triebel lizorkin sobolev spaces dominating mixed smoothness considered

Affiliations des auteurs :

Lev Markhasin ¹

¹ Institut für Stochastik und Anwendungen Universität Stuttgart Pfaffenwaldring 57 70569 Stuttgart, Germany

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Lev Markhasin. $L_p$- and $S_{p,q}^rB$-discrepancy of (order $2$) digital nets. Acta Arithmetica, Tome 168 (2015) no. 2, pp. 139-159. doi : 10.4064/aa168-2-4. http://geodesic.mathdoc.fr/articles/10.4064/aa168-2-4/

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