Discrepancy of Hammersley points in Besov spaces of dominating mixed smoothness

Abstract

AbstractWe 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)