Approximation of Quasi-Monte Carlo worst case error in weighted spaces of infinitely times smooth functions

TL;DR

This paper analyzes the worst case error of Quasi-Monte Carlo methods for smooth functions, showing that digital nets with low exponential function error also perform well for the entire function space.

Contribution

It establishes bounds linking QMC worst case error to exponential function errors, simplifying error estimation for smooth function classes.

Findings

Bounded ratio between worst case error and exponential function error

Digital nets with low exponential error yield small overall integration error

Provides a simple interpretation of QMC error behavior in smooth spaces

Abstract

In this paper, we consider Quasi-Monte Carlo (QMC) worst case error of weighted smooth function classes in $C^\infty[0,1]^s$ by a digital net over $\mathbb F_2$. We show that the ratio of the worst case error to the QMC integration error of an exponential function is bounded above and below by constants. This result provides us with a simple interpretation that a digital net with small QMC integration error for an exponential function also gives the small integration error for any function in this function space.