General discrepancy estimates III: The Erd�s-Tur�n-Koksma inequality for the Haar function system

Abstract

The classical Erdos-Turan-Koksma inequality gives us an upper bound for the discrepancy of a sequence in thes-dimensional unit cube in terms of exponential sums, more precisely, in terms of the trigonometric function system. In this paper, we shall prove the inequality of Erdos-Turan-Koksma for the extreme and the star discrepancy, for generalized Haar function systems. Further, we shall show the existence of the inequality of Erdos-Turan-Koksma for the isotropic discrepancy, for generalized Haar and Walsh function systems.