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Original Paper
Discrepancy of Hammersley points in Besov spaces of dominating mixed smoothness
Article first published online: 26 FEB 2010
DOI: 10.1002/mana.200910265
Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Issue
1522-2616/asset/cover.gif?v=1&s=0110fef2c1f789362b0c8808bc8248840b71170f "Mathematische Nachrichten")
Mathematische Nachrichten
Special Issue: Erhard Schmidt Memorial Issue, Part III
Volume 283, Issue 3, pages 478–488, March 2010
Additional Information
How to Cite
Hinrichs, A. (2010), Discrepancy of Hammersley points in Besov spaces of dominating mixed smoothness. Math. Nachr., 283: 478–488. doi: 10.1002/mana.200910265
Publication History
- Issue published online: 26 FEB 2010
- Article first published online: 26 FEB 2010
- Manuscript Accepted: 26 DEC 2009
- Manuscript Received: 12 DEC 2009
- Abstract
- References
- Cited By
Keywords:
- Discrepancy;
- Hammersley point set;
- Besov spaces;
- dominating mixed smoothness;
- quasi-Monte Carlo;
- numerical integration
Abstract
We study the discrepancy function of two-dimensional Hammersley type point sets in the unit square. It is well-known that the symmetrized Hammersley point set achieves the asymptotically best possible rate for the L2-norm of the discrepancy function. In this paper we consider the norm of the discrepancy function of Ham¬mersley type point sets in Besov spaces of dominating mixed smoothness and show that it achieves the optimal rate under appropriate assumptions on the set and the smoothness parameter of the Besov space. Our proof relies on a characterization of the Besov spaces of dominating mixed smoothness via coefficients in the Haar expansion and the computation of the Haar expansion of the discrepancy function of Hammersley type point sets (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)