The curse of dimensionality for the Lp-discrepancy with finite p

Abstract

The Lp-discrepancy is a quantitative measure for the irregularity of distribution of an N-element point set in the d-dimensional unit-cube, which is closely related to the worst-case error of quasi-Monte Carlo algorithms for numerical integration. It's inverse for dimension d and error threshold ε∈(0,1) is the minimal number of points in [0,1)d such that the minimal normalized Lp-discrepancy is less or equal ε. It is well known, that the inverse of L2-discrepancy grows exponentially fast with the dimension d, i.e., we have the curse of dimensionality, whereas the inverse of L∞-discrepancy depends exactly linearly on d. The behavior of inverse of Lp-discrepancy for general p∉{2,∞} has been an open problem for many years. In this paper we show that the Lp-discrepancy suffers from the curse of dimensionality for all p in (1,2] which are of the form p=2ℓ/(2ℓ−1) with ℓ∈N.This result follows from a more general result that we show for the worst-case error of numerical integration in an anchored Sobolev space with anchor 0 of once differentiable functions in each variable whose first derivative has finite Lq-norm, where q is an even positive integer satisfying 1/p+1/q=1.