TY - JOUR
T1 - Discrepancy for randomized riemann sums
AU - Brandolini, Luca
AU - Chen, William
AU - Gigante, Giacomo
AU - Travaglini, Giancarlo
N1 - Copyright 2009 American Mathematical Society. First published in Proceedings of the American Mathematical Society, Volume 137, Issue 10, pp. 3187-3196, published by the American Mathematical Society. The original article can be found at http://dx.doi.org/10.1090/S0002-9939-09-09975-4.
PY - 2009/10
Y1 - 2009/10
N2 - Given a finite sequence UN = {u1,., uN} of points contained in the d-dimensional unit torus, we consider the L2 discrepancy between the integral of a given function and the Riemann sums with respect to translations of UN. We show that with positive probability, the L2 discrepancy of other sequences close to UN in a certain sense preserves the order of decay of the discrepancy of UN. We also study the role of the regularity of the given function.
AB - Given a finite sequence UN = {u1,., uN} of points contained in the d-dimensional unit torus, we consider the L2 discrepancy between the integral of a given function and the Riemann sums with respect to translations of UN. We show that with positive probability, the L2 discrepancy of other sequences close to UN in a certain sense preserves the order of decay of the discrepancy of UN. We also study the role of the regularity of the given function.
UR - http://www.scopus.com/inward/record.url?scp=77951049048&partnerID=8YFLogxK
U2 - 10.1090/S0002-9939-09-09975-4
DO - 10.1090/S0002-9939-09-09975-4
M3 - Article
AN - SCOPUS:77951049048
SN - 0002-9939
VL - 137
SP - 3187
EP - 3196
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 10
ER -