Bounds for the weighted L^p discrepancy and tractability of integration

Abstract

Quite recently Sloan and Woźniakowski (J. Complexity 14 (1998) 1) introduced a new notion of discrepancy, the so-called weighted L^p 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^p discrepancy for even p. We use this formula to derive an upper bound for the average weighted L^p 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 L^p 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.