The classical Koksma-Hlawka inequality does not apply to functions with simple discontinuities. Here we state a Koksma-Hlawka type inequality which applies to piecewise smooth functions fχΩ, with f smooth and Ω a Borel subset of [0,1]d: |N-1∑j=1N(f χΩ)(xj)-∫Ωf(x) dx|≤D(Ω,(xj)j=1N)V(f), where D(Ω,(xj)j=1N) is the discrepancy D(Ω,(xj)j=1N)=2dsupI E[0,1]d(|N-1∑j= 1NχΩ∩I(xj)-|Ω∩I||), the supremum is over all d-dimensional intervals, and V(f) is the total variation V(f)=∑α(0,1)d2d-|α|∫ javax.xml.bind.JAXBElement@12009206|(∂∂x)αf(x)|dx. We state similar results with variation and discrepancy measured by Lp and Lq norms, 1/p+1/q=1, and we also give extensions to compact manifolds.
Brandolini, L., Colzani, L., Gigante, G., Travaglini, G. (2013). On the Koksma-Hlawka Inequality. JOURNAL OF COMPLEXITY, 29(2), 158-172 [10.1016/j.jco.2012.10.003].
On the Koksma-Hlawka Inequality
COLZANI, LEONARDO;TRAVAGLINI, GIANCARLO
2013
Abstract
The classical Koksma-Hlawka inequality does not apply to functions with simple discontinuities. Here we state a Koksma-Hlawka type inequality which applies to piecewise smooth functions fχΩ, with f smooth and Ω a Borel subset of [0,1]d: |N-1∑j=1N(f χΩ)(xj)-∫Ωf(x) dx|≤D(Ω,(xj)j=1N)V(f), where D(Ω,(xj)j=1N) is the discrepancy D(Ω,(xj)j=1N)=2dsupI E[0,1]d(|N-1∑j= 1NχΩ∩I(xj)-|Ω∩I||), the supremum is over all d-dimensional intervals, and V(f) is the total variation V(f)=∑α(0,1)d2d-|α|∫ javax.xml.bind.JAXBElement@12009206|(∂∂x)αf(x)|dx. We state similar results with variation and discrepancy measured by Lp and Lq norms, 1/p+1/q=1, and we also give extensions to compact manifolds.
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