Optimal cubature in Besov spaces with dominating mixed smoothness on the unit square

Abstract

We prove new optimal bounds for the error of numerical integration in bivariate Besov spaces with dominating mixed order r. The results essentially improve on the so far best known upper bound achieved by using cubature formulas taking points from a sparse grid. Motivated by Hinrichs’ observation that Hammersley type point sets provide optimal discrepancy estimates in Besov spaces with mixed smoothness on the unit square, we directly study quasi-Monte Carlo integration on such point sets. As the main tool we prove the representation of a bivariate periodic function in a piecewise linear tensor Faber basis. This allows for optimal worst case estimates of the QMC integration error with respect to Besov spaces with dominating mixed smoothness up to order r<2. The results in this paper are the first step towards sharp results for spaces with arbitrarily large mixed order on the d-dimensional unit cube. In fact, in contrast to Fibonacci lattice rules, which are also practicable in this context, the QMC methods used in this paper have a proper counterpart in d dimensions.