On the Koksma–Hlawka inequality

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 fχΩfχΩ, with ff smooth and ΩΩ a Borel subset of [0,1]d[0,1]d: |N−1∑j=1N(fχΩ)(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⊆[0,1]d{|N−1∑j=1NχΩ∩I(xj)−|Ω∩I||}, the supremum is over all dd-dimensional intervals, and V(f)V(f) is the total variation V(f)=∑α∈{0,1}d2d−|α|∫[0,1]d|(∂∂x)αf(x)|dx. We state similar results with variation and discrepancy measured by LpLp and LqLq norms, 1/p+1/q=11/p+1/q=1, and we also give extensions to compact manifolds.