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

DOI:
https://doi.org/10.1090/S0002-9939-09-09975-4

Published electronically:
May 19, 2009

MathSciNet review:
2515389

Full-text PDF

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.

**1.**J. Beck,*Irregularities of distribution. I*, Acta Math.**159**(1987), 1-49. MR**906524 (89c:11117)****2.**J. Beck and W.W.L. Chen,*Irregularities of Distribution*, Cambridge Tracts in Mathematics, vol. 89, Cambridge University Press, 1987. MR**903025 (88m:11061)****3.**J. Beck and W.W.L. Chen,*Note on irregularities of distribution. II*, Proc. London Math. Soc. (3)**61**(1990), 251-272. MR**1063047 (91g:11083)****4.**L. Brandolini, L. Colzani, and G. Travaglini,*Average decay of Fourier transforms and integer points in polyhedra*, Ark. Mat.**35**(1997), 253-275. MR**1478780 (99e:42021)****5.**L. Brandolini, M. Rigoli, and G. Travaglini,*Average decay of Fourier transforms and geometry of convex sets*, Rev. Mat. Iberoamericana**14**(1998), 519-560. MR**1681584 (2000a:42017)****6.**W.W.L. Chen and G. Travaglini,*Deterministic and probabilistic discrepancies*, Ark. Mat. (to appear).**7.**C.S. Herz,*Fourier transforms related to convex sets*, Ann. of Math. (2)**75**(1962), 81-92. MR**0142978 (26:545)****8.**F.J. Hickernell and H. Woźniakowski,*The price of pessimism for multidimensional quadrature*, J. Complexity**17**(2001), 625-659. MR**1881662 (2002m:60013)****9.**D.P. Mitchell,*Consequences of stratified samplings in graphics*, SIGGRAPH 96: Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques, Association for Computing Machinery, 1996, pp. 277-280.**10.**H.L. Montgomery,*Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis*, CBMS Regional Conference Series in Mathematics, vol. 84, Amer. Math. Soc., Providence, RI, 1994. MR**1297543 (96i:11002)****11.**S.M. Nikol'skiı,*Approximation of Functions of Several Variables and Imbedding Theorems*, Springer-Verlag, New York-Heidelberg, 1975. MR**0374877 (51:11073)****12.**L. Parnovski and A.V. Sobolev,*On the Bethe-Sommerfeld conjecture for the polyharmonic operator*, Duke Math. J.**107**(2001), 209-238. MR**1823047 (2002d:35050)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
11K38,
41A55

Retrieve articles in all journals with MSC (2000): 11K38, 41A55

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:
https://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.