Low-discrepancy sequences for piecewise smooth functions on the two-dimensional torus

We produce explicit low-discrepancy infinite sequences which can be used to approximate the integral of a smooth periodic function restricted to a convex domain with positive curvature in R2. The proof depends on simultaneous diophantine approximation and a general version of the Erdős–Turán inequal...

Full description

Saved in:
Bibliographic Details
Published in:Journal of Complexity 2016-04, Vol.33, p.1-13
Main Authors: Brandolini, Luca, Colzani, Leonardo, Gigante, Giacomo, Travaglini, Giancarlo
Format: Article
Language:English
Subjects:
Approximation
Curvature
Diophantine approximation
Discrepancy
Erdős–Turán inequality
Inequalities
Integrals
Koksma–Hlawka inequality
Mathematical analysis
Periodic functions
Piecewise smooth functions
Sequences
Two dimensional
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We produce explicit low-discrepancy infinite sequences which can be used to approximate the integral of a smooth periodic function restricted to a convex domain with positive curvature in R2. The proof depends on simultaneous diophantine approximation and a general version of the Erdős–Turán inequality. •We present sequences in the plane with low discrepancy.•The discrepancy is with respect to a smooth convex set intersected with rectangles.•This allows to numerically approximate integrals of piecewise smooth functions.•Thanks to a general Erdős–Turán inequality we avoid using isotropic discrepancy.•The construction of the sequence is based on simultaneous diophantine approximation.
ISSN:0885-064X
1090-2708
DOI:10.1016/j.jco.2015.09.003