Tent transformed order 2 nets and quasi-Monte Carlo rules with quadratic error decay

We investigate quasi-Monte Carlo quadrature of non-periodic functions with bounded mixed second derivative over the cube. By using the so-called tent transform and its mapping properties we inherit error bounds for the performance of order 2 2 digital nets and lattices from the periodic setting. Our analysis is based on decay properties of the multivariate Faber-Schauder coefficients of functions with bounded mixed second weak derivatives. As already observed by Hinrichs, Markhasin, Oettershagen, and T. Ullrich [Numer. Math. 134 (2016), pp. 163–196], order 2 2 nets work particularly well on tensorized (periodic) Faber splines. From this we obtain a quadratic decay rate for tent transformed order 2 2 nets also in the nonperiodic setting. This improves the formerly best known bound for this class of point sets by a factor of log ⁡ N \log N . In addition, we show the quadratic decay of multivariate Fourier coefficients of tent transformed functions with second order bounded mixed weak derivatives on the cube. This allows for the use of tent transformed lattice rules in this context. We back up our findings with numerical experiments, suggesting that the bounds for order 2 2 nets can be improved even further. This particularly indicates that point sets of lower complexity (compared to previously considered constructions) may already give (near) optimal error decay rates for quadrature of functions with second order mixed smoothness.