Discrepancy and integration in function spaces with dominating mixed smoothness

Lev Markhasin. Discrepancy and integration in function spaces with dominating mixed smoothness. 2013. <http://eudml.org/doc/285956>.

@book{LevMarkhasin2013,
abstract = {Optimal lower bounds for discrepancy in Besov spaces with dominating mixed smoothness are known from the work of Triebel. Hinrichs proved upper bounds in the plane. In this work we systematically analyse the problem, starting with a survey of discrepancy results and the calculation of the best known constant in Roth's Theorem. We give a larger class of point sets satisfying the optimal upper bounds than already known from Hinrichs for the plane and solve the problem in arbitrary dimension for certain parameters considering celebrated constructions by Chen and Skriganov which are known to achieve the optimal L₂-norm of the discrepancy function. Since those constructions are b-adic, we give b-adic characterizations of the spaces. Finally results for Triebel-Lizorkin and Sobolev spaces with dominating mixed smoothness and for the integration error are concluded.},
author = {Lev Markhasin},
keywords = {discrepancy; numerical integration; quasi-Monto Carlo algorithms; Haar system; Hammersley point sets; Chen-Skriganov point sets; dominating mixed smoothness},
language = {eng},
title = {Discrepancy and integration in function spaces with dominating mixed smoothness},
url = {http://eudml.org/doc/285956},
year = {2013},
}

TY - BOOK
AU - Lev Markhasin
TI - Discrepancy and integration in function spaces with dominating mixed smoothness
PY - 2013
AB - Optimal lower bounds for discrepancy in Besov spaces with dominating mixed smoothness are known from the work of Triebel. Hinrichs proved upper bounds in the plane. In this work we systematically analyse the problem, starting with a survey of discrepancy results and the calculation of the best known constant in Roth's Theorem. We give a larger class of point sets satisfying the optimal upper bounds than already known from Hinrichs for the plane and solve the problem in arbitrary dimension for certain parameters considering celebrated constructions by Chen and Skriganov which are known to achieve the optimal L₂-norm of the discrepancy function. Since those constructions are b-adic, we give b-adic characterizations of the spaces. Finally results for Triebel-Lizorkin and Sobolev spaces with dominating mixed smoothness and for the integration error are concluded.
LA - eng
KW - discrepancy; numerical integration; quasi-Monto Carlo algorithms; Haar system; Hammersley point sets; Chen-Skriganov point sets; dominating mixed smoothness
UR - http://eudml.org/doc/285956
ER -