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Discrepancy of Hammersley points in Besov spaces of dominating mixed smoothness
Author(s) -
Hinrichs Aicke
Publication year - 2010
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200910265
Subject(s) - mathematics , smoothness , besov space , norm (philosophy) , haar , function (biology) , pure mathematics , mathematical analysis , wavelet , interpolation space , computer science , biochemistry , chemistry , functional analysis , artificial intelligence , evolutionary biology , biology , political science , law , gene
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 L 2 ‐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)