Bounds for the weighted Lp discrepancy and tractability of integration

AbstractQuite recently Sloan and Woźniakowski (J. Complexity 14 (1998) 1) introduced a new notion of discrepancy, the so-called weighted Lp 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 Lp discrepancy for even p. We use this formula to derive an upper bound for the average weighted Lp discrepancy. This bound enables us to give conditions on the sequence of weights γ such that there exists N points in [0,1)d for which the weighted Lp discrepancy is uniformly bounded in d and goes to zero polynomially in N−1.Finally we use these facts to generalize some results from Sloan and Woźniakowski (1998) on (strong) QMC-tractability of integration in weighted Sobolev spaces