Discrepancy of Hammersley points in Besov spaces of dominating mixed smoothness

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 Hammersley 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.

Lp- and Sp,qrB-discrepancy of (order 2) digital nets

“…Hinrichs [Hi10] proved for d = 2 that for all 1 ≤ p, q ≤ ∞ and all 0 ≤ r < 1/p there exists a constant C p,q,r > 0 such that for every integer N ≥ 2 there exists a point set P in [0, 1) 2 with N points such that…”

“…The notion S r p,q B-discrepancy will be defined in the next section. In d = 2 also (generalized) Hammersley point sets can be used (see [Hi10], [M13a]). Our goal is to prove that there are also other point sets with optimal bounds on the S r p,q Bdiscrepancy.…”

On lower bounds for the L2-discrepancy

“…A convenient method to do this is to compute the Fourier coefficients with respect to the Haar system and then use Parseval's formula. This method has already been used in a recent paper [4] by the first author to prove optimal upper estimates for the discrepancy of Hammersley-type point sets measured in spaces of dominating mixed smoothness.…”

“…Constructions of point sets satisfying (4) are plenty, the first one was given by Davenport [1]. For further constructions and the general theory of discrepancy, we refer the reader to the books [2,[6][7][8].…”

Change of Variable in Spaces of Mixed Smoothness and Numerical Integration of Multivariate Functions on the Unit Cube

“…This research area developed into several directions and attracted a lot of interest in the past 50 years, starting with the seminal papers by Korobov [20], Hlawka [18], and Bakhvalov [2]. Afterwards many authors contributed to the construction and analysis of optimal cubature formulae for multivariate functions (with bounded mixed derivative), see, e.g., Frolov [12], Bykovskii [3], Temlyakov [33][34][35][36][37][38][39], Dubinin [7,8], Skriganov [31], Triebel [43], Hinrichs et al [16,17], Markhasin [23], Novak and Woźniakowski [26], Krieg and Novak [21], Dick and Pillichshammer [6], Dũng and Ullrich [10], Goda et al [14,15], and Ullrich [45], to mention just a few. More historical comments and further references can be found at the end of this introduction as well as in the recent survey paper [9,Sect.…”