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

作者：Josef Dick · 发表于：SIAM Journal on Numerical Analysis · 年份：2008 · DOI：10.1137/060666639 · 被引用次数：139 · 研究领域：Mathematical Approximation and Integration、Digital Image Processing Techniques、Analytic Number Theory Research

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.