Discretization of integrals on compact metric measure spaces

Abstract

Let μ be a Borel probability measure on a compact path-connected metric space (X,ρ) for which there exist constants c,β≥1 such that μ(B)≥crβ for every open ball B⊂X of radius r>0. For a class of Lipschitz functions Φ:[0,∞)→R that are piecewise within a finite-dimensional subspace of continuous functions, we prove under certain mild conditions on the metric ρ and the measure μ that for each positive integer N≥2, and each g∈L∞(X,dμ) with ‖g‖∞=1, there exist points y1,…,yN∈X and real numbers λ1,…,λN such that for any x∈X,|∫XΦ(ρ(x,y))g(y)dμ(y)−∑j=1NλjΦ(ρ(x,yj))|⩽CN−12−32βlog⁡N, where the constant C>0 is independent of N and g. In the case when X is the unit sphere Sd of Rd+1 with the usual geodesic distance, we also prove that the constant C here is independent of the dimension d. Our estimates are better than those obtained from the standard Monte Carlo methods, which typically yield a weaker upper bound N−12log⁡N.