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

TL;DR

This paper introduces Walsh spaces that include smooth functions with finite derivatives, and demonstrates that digital quasi-Monte Carlo rules based on specific sequences achieve optimal convergence rates for numerical integration of these functions.

Contribution

It defines a new Walsh space encompassing smooth functions and proves that digital $(t,eta,s)$-sequences attain optimal convergence rates for integration within this space.

Findings

Digital $(t,eta,s)$-sequences achieve optimal convergence rates.

Walsh spaces include Sobolev spaces with smooth functions.

Explicit constructions of sequences are provided.

Abstract

We define a Walsh space which contains all functions whose partial mixed derivatives up to order $\delta \ge 1$ 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 $(t,\alpha,s)$-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 $(t,\alpha,s)$-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.