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In a recent paper by the authors, it is shown that there exists a quasi-Monte Carlo (QMC) rule which achieves the best possible rate of convergence for numerical integration in a reproducing kernel Hilbert space consisting of smooth functions.
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J. Dick, Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions, SIAM J. Numer. Anal. 45 (2007), 2141–2176
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J. Dick, Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order, SIAM J. Numer. Anal. 46 (2008), 1519–1553
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J. Baldeaux and J. Dick, QMC rules of arbitrary high order: Reproducing kernel Hilbert space approach, Constr. Approx. 30 (2009), 495–527
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T. Goda, K. Suzuki and T. Yoshiki, The b b -adic tent transformation for quasi-Monte Carlo integration using digital nets, J. Approx. Theory 194 (2015), 62–86
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T. Goda, K. Suzuki and T. Yoshiki, Digital nets with infinite digit expansions and construction of folded digital nets for quasi-Monte Carlo integration, J. Complexity (2015) DOI:10.1016/j.jco.2015.09.005
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A. Hinrichs, L. Markhasin, J. Oettershagen and T. Ullrich, Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions, Numer. Math. (2015) DOI:10.1007/s00211-015-0765-y
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K. Suzuki and T. Yoshiki, Formulas for the Walsh coefficients of smooth functions and their application to bounds on the Walsh coefficients, J. Approx. Theory (2016), DOI:10.1016/j.jat.2015.12.002
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