The role of Frolov's cubature formula for functions with bounded mixed derivative

We prove upper bounds on the order of convergence of Frolov's cubature formula for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study worst-case integration errors for Besov $\mathbf{B}^s_{p,\theta}$ and Triebel-Lizorkin spaces $\mathbf{F}^s_{p,\theta}$ and our results treat the whole range of admissible parameters $(s\geq 1/p)$. In particular, we obtain upper bounds for the difficult the case of small smoothness which is given for Triebel-Lizorkin spaces $\mathbf{F}^s_{p,\theta}$ in case $1<\theta<p<\infty$ with $1/p<s\leq 1/\theta$. The presented upper bounds on the worst-case error show a completely different behavior compared to "large" smoothness $s>1/\theta$. In the latter case the presented upper bounds are optimal, i.e., they can not be improved by any other cubature formula. The optimality for "small" smoothness is open.