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

In this paper, we consider Quasi-Monte Carlo (QMC) worst case error of weighted smooth function classes in $C^{\mathrm{\infty }}[0,1{]}^s$ 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.