Super-polynomial convergence and tractability of multivariate integration for infinitely times differentiable functions

Abstract

We investigate multivariate integration for a space of infinitely times differentiable functions Fs,u:={f∈C∞[0,1]s∣‖f‖Fs,u<∞}, where ‖f‖Fs,u:=supα=(α1,…,αs)∈N0s‖f(α)‖L1/∏j=1sujαj, f(α):=∂∣α∣∂x1α1⋯∂xsαsf and u={uj}j≥1 is a sequence of positive decreasing weights. Let e(n,s) be the minimal worst-case error of all algorithms that use n function values in the s-variate case. We prove that for any u and s considered e(n,s)≤C(s)exp(−c(s)(logn)2) holds for all n, where C(s) and c(s) are constants which may depend on s. Further we show that if the weights u decay sufficiently fast then there exist some 1<p<2 and absolute constants C and c such that e(n,s)≤Cexp(−c(logn)p) holds for all s and n. These bounds are attained by quasi-Monte Carlo integration using digital nets. These convergence and tractability results come from those for the Walsh space into which Fs,u is embedded.