We prove a correspondence principle between multivariate functions of bounded variation in the sense of Hardy and Krause and signed measures of finite total variation, which allows us to obtain a simple proof of a generalized Koksma-Hlawka inequality for non-uniform measures. Applications of this inequality to importance sampling in Quasi-Monte Carlo integration and tractability theory are given. We also discuss the problem of transforming a low-discrepancy sequence with respect to the uniform measure into a sequence with low discrepancy with respect to a general measure μ, and show the limitations of a method suggested by Chelson.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa167-2-4,
author = {Christoph Aistleitner and Josef Dick},
title = {Functions of bounded variation, signed measures, and a general Koksma-Hlawka inequality},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {143-171},
zbl = {1326.11038},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa167-2-4}
}
Christoph Aistleitner; Josef Dick. Functions of bounded variation, signed measures, and a general Koksma-Hlawka inequality. Acta Arithmetica, Tome 168 (2015) pp. 143-171. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa167-2-4/