Walsh Spaces Containing Smooth Functions and Quasi-Monte Carlo Rules of Arbitrary High Order

We define a Walsh space which contains all functions whose partial mixed derivatives up to order δ ≥ 1 exist and have finite variation. In particular, for a suitable choice of parameters, this implies that certain reproducing kernel Sobolev spaces are contained in these Walsh spaces. For this Walsh space we then show that quasi-Monte Carlo rules based on digital (t, α, s)-sequences achieve the optimal rate of convergence of the worst-case error for numerical integration. This rate of convergence is also optimal for the subspace of smooth functions. Explicit constructions of digital (t, α, s)sequences are given hence providing explicit quasi-Monte Carlo rules which achieve the optimal rate of convergence of the integration error for arbitrarily smooth functions.