Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order

Abstract

We define a Walsh space which contains all functions whose partial mixed derivatives up to order $δ\ge 1$ exist and have finite variation. In particular, for a suitable choice of parameters, this implies that certain 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.