Finding exact formulas for the L_2 discrepancy of digital (0,n,2)-nets via Haar functions [article]

Ralph Kritzinger
2017 arXiv   pre-print
We use the Haar function system in order to study the L_2 discrepancy of a class of digital (0,n,2)-nets. Our approach yields exact formulas for this quantity, which measures the irregularities of distribution of a set of points in the unit interval. We will obtain such formulas not only for the classical digital nets, but also for shifted and symmetrized versions thereof. The basic idea of our proofs is to calculate all Haar coefficents of the discrepancy function exactly and insert them into
more » ... arseval's identity. We will also discuss reasons why certain (symmetrized) digital nets fail to achieve the optimal order of L_2 discrepancy and use the Littlewood-Paley inequality in order to obtain results on the L_p discrepancy for all p∈ (1,∞).
arXiv:1711.06058v1 fatcat:6hjapaoccrgxjhzrj3v3unt32y