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 | References | Similar Articles | Additional Information

Abstract: Given a finite sequence of points contained in the -dimensional unit torus, we consider the discrepancy between the integral of a given function and the Riemann sums with respect to translations of . We show that with positive probability, the discrepancy of other sequences close to in a certain sense preserves the order of decay of the discrepancy of . 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

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.