By Klin M., et al.

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**Example text**

4 On Applications and Connections 25 ° for a parameter r 2:: and for a point y E [O,l]d, define a function hy by setting hy(x) = n~=l max(O,xk - YkY, and let :F = {hy: Y E [O,l]d}. The resulting notion of discrepancy is called the r-smooth discrepancy. e. supplies good error bounds for numerical integration of a possibly much wider class of functions. A modern approach to this issue uses the so-called reproducing kernels in Hilbert spaces of functions; a little more on this can be found in the remarks below.

Thus, we approximate the continuous measure voID by a (signed) measure concentrated on an n-point set. Actually, there are at least four different notions of discrepancy involving weighted point sets: we may require the weights to be nonnegative and to sum up to n = IPI, or we may drop one of these two conditions or both (negative weights are not as absurd as it might seem, since some of the classical quadrature formulas, such as the Newton-Cotes rule, involve negative coefficients). For discrepancy theory, the generalization to weighted point sets is usually not too significant-most of the lower bounds in this book, say, go through for weighted point sets without much difficulty.

By the Chinese remainder theorem, the set {O, 1, 2, ... ) Therefore, the points of P lying in S are evenly spaced in the x-direction with step 2;~. The box S can again be divided into boxes of length 2;~ in the x-direction, and each of these boxes has zero discrepancy. Similar to the planar case, it follows that the discrepancy of the box [0, a) x I x J considered in Claim I is at most 1. Claim II. Any corner C(x,y,z) can be expressed as a disjoint union of at most flog2 n1. 2 flOg3 n1 boxes as in Claim I, plus a set M c [0, 1P with ID(P,M)I :::; 2.