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arXiv · 1304.0328

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

Josef Dick
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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.

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Josef Dick. 2013-04-01. Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. https://doi.org/10.1137/060666639

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