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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Discrepancy for randomized Riemann sums


Authors: Luca Brandolini, William Chen, Giacomo Gigante and Giancarlo Travaglini
Journal: Proc. Amer. Math. Soc. 137 (2009), 3187-3196
MSC (2000): Primary 11K38; Secondary 41A55
Published electronically: May 19, 2009
MathSciNet review: 2515389
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Abstract: Given a finite sequence $ U_{N}=\{u_{1},\ldots,u_{N}\}$ of points contained in the $ d$-dimensional unit torus, we consider the $ L^{2}$ discrepancy between the integral of a given function and the Riemann sums with respect to translations of $ U_{N}$. We show that with positive probability, the $ L^{2}$ discrepancy of other sequences close to $ U_{N}$ in a certain sense preserves the order of decay of the discrepancy of $ U_{N}$. We also study the role of the regularity of the given function.


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Additional Information

Luca Brandolini
Affiliation: Dipartimento di Ingegneria dell’Informazione e Metodi Matematici, Università di Bergamo, Viale Marconi 5, 24044 Dalmine, Bergamo, Italy
Email: luca.brandolini@unibg.it

William Chen
Affiliation: Department of Mathematics, Macquarie University, Sydney, NSW 2109, Australia
Email: wchen@maths.mq.edu.au

Giacomo Gigante
Affiliation: Dipartimento di Ingegneria dell’Informazione e Metodi Matematici, Università di Bergamo, Viale Marconi 5, 24044 Dalmine, Bergamo, Italy
Email: giacomo.gigante@unibg.it

Giancarlo Travaglini
Affiliation: Dipartimento di Statistica, Edificio U7, Università di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy
Email: giancarlo.travaglini@unimib.it

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09975-4
PII: S 0002-9939(09)09975-4
Keywords: Irregularities of distribution, decay of Fourier coefficients
Received by editor(s): June 24, 2008
Published electronically: May 19, 2009
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.