Discrepancy for randomized Riemann sums

Brandolini, Luca; Chen, William; Gigante, Giacomo; Travaglini, Giancarlo

doi:10.1090/S0002-9939-09-09975-4

Discrepancy for randomized Riemann sums
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by Luca Brandolini, William Chen, Giacomo Gigante and Giancarlo Travaglini PDF

Proc. Amer. Math. Soc. 137 (2009), 3187-3196 Request permission

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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