Bounds for the weighted Lp discrepancy and tractability of integration

Quite recently Sloan and Woźniakowski [4] introduced a new notion of discrepancy, the so called weighted L discrepancy of points in the d-dimensional unit cube for a sequence γ = (γ1, γ2, . . .) of weights. In this paper we prove a nice formula for the weighted L discrepancy for even p. We use this formula to derive an upper bound for the average weighted L discrepancy. This bound enables us to give conditions on the sequence of weights γ such that there exists N points in [0, 1) for which the weighted L discrepancy is uniformly bounded in d and goes to zero polynomially in N. Finally we use these facts to generalize some results from Sloan and Woźniakowski [4] on (strong) QMC-tractability of integration in weighted Sobolev spaces.