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

Abstract

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 Parseval'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, ∞).

Point sets with optimal order of extreme and periodic discrepancy

“…For the latter it even suffices to evaluate only those coefficients where j ∈ N d 0 . The Haar coefficients of the discrepancy function of certain digital nets have been computed exactly in [19]. We use these results to obtain exact formulas for the extreme L 2 discrepancy of these nets.…”

“…Since the Haar coefficients of the respective discrepancy functions have already been computed, there is not much left to do. Just add the expressions given in [19, to obtain the result for (L extr 2,2 m (P a (σ))) 2 . Caution: before applying the results from [19, here, they have to be multiplied with 2 2m since in [19] a normalized version of the discrepancy function is considered.…”

“…Just add the expressions given in [19, to obtain the result for (L extr 2,2 m (P a (σ))) 2 . Caution: before applying the results from [19, here, they have to be multiplied with 2 2m since in [19] a normalized version of the discrepancy function is considered.…”

Digital nets in dimension two with the optimal order of L_p discrepancy

“…The rest follows with (9). ✷ For the following two propositions, we use the shorthand R = r 1 ⊕ · · · ⊕ r j 1 .…”

“…Hence the difference of these two expressions is given by 1 t 1 ,...,t n−j 1 −1 =0 uv(1) − 1 t 1 ,...,t n−j 1 −1 =0 uv(0) = 1 4 a n−j 1 −1 (2R − 1). Now we put everything together and use (9) to find the claimed result on the Haar coefficients. ✷ Case 6: j ∈ J 6 := {(j 1 , j 2 ) : j 1 + j 2 ≤ n − 3} Proposition 6 Let j ∈ J 6 and m ∈ D j .…”