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 [email protected]"@W, with f smooth and @W a Borel subset of [0,1]^d: |N^-^[email protected]?j=1N([email protected]"@W)(x"j)[email protected]!"@Wf(x)dx|@?D(@W,{x"j}"j"="1^N)V(f), where D(@W,{x"j}"j"="1^N) is the discrepancy D(@W,{x"j}"j"="1^N)=2^[email protected]?[0,1]^d{|N^-^[email protected][email protected]"@W"@?"I(x"j)-|@[email protected]?I||}, the supremum is over all d-dimensional intervals, and V(f) is the total variation V(f)[email protected][email protected]@?{0,1}^d2^d^-^|^@a^|@!"["0","1"]"^"d|(@[email protected]?x)^@af(x)|dx. We state similar results with variation and discrepancy measured by L^p and L^q norms, 1/p+1/q=1, and we also give extensions to compact manifolds.