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We investigate quasi-Monte Carlo integration using higher order digital nets in weighted Sobolev spaces of arbitrary fixed smoothness $\alpha \in \mathbb{N}$, $\alpha \ge 2$, defined over the $s$-dimensional unit cube.
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J. Dick and F. Pillichshammer, Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration , Cambridge University Press, Cambridge, 2010
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J. Baldeaux, J. Dick and F. Pillichshammer, Duality theory and propagation rules for higher order nets, Discrete Math., 311 (2011) 362–386
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L. Markhasin, Quasi-Monte Carlo methods for integration of functions with dominating mixed smoothness in arbitrary dimension, J. Complexity, 29 (2013) 370–388
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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. Dick, D. Nuyens and F. Pillichshammer, Lattice rules for nonperiodic smooth integrands, Numer. Math., 126 (2014) 259–291
2014
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