Niederreiter Theorem 2.12

Niederriether Theorem 2.12 (his numbering) proves that the Koksma-Hlawka inequality is the best possible inequality when the function h may come from a broad set of functions. The function space he considers is the set of functions with Hardy-Krause variation equal to 1 and which have continuous derivatives of all orders. The functions can depend on s dimensions and are defined on the unit hypercube in s dimensions. The value of the function is a real number.

Theorem 5 (No go theorem to beat Koksma-Hlawka)   For any points $x_1,...,x_N$ in the unit hypercube in s dimensions, and any $\epsilon>0$ there exists a function h that has Hardy-Krause variation equal to 1 on the unit hyper cube in s dimensions, and which has continuous derivatives of all orders such that

$$\vert\frac{1}{N} \sum_{i=1}^{N} h(x_i) - \int h(u) du > V(h) D_{N}^{*}(x_1,...,x_N) - \epsilon$$ (2.14)

where V(h) is the Hardy-Krause Variation in s dimensions of the function h on the unit hypercube in s dimensions. Here the integral is over the unit hypercube in s dimensions.

We left in the V(h) even though it is 1 to emphasize the result.

What this theorem says is that there is no tighter bound than Koksma-Hlawka when h can be any function in the allowed function space (set of functions). This theorem of Niederreiter and the Koksma-Hlawka theorem show that Discrepancy is a basic part of the theory of integration that within certain specifications can't be improved upon.