Kriterien in der Theorie der Gleichverteilung

It is the aim of this paper to introduce two new notions of discrepancy. They are defined by the formulas $$begin{gathered} Delta _N^r left( {omega ;f} right) = mathop {sup }limits_{left| z right| = r} left| {left( {{1 mathord{left/ {vphantom {1 N}} right. kern-nulldelimiterspace} N}} right)sumlimits_{n = 1}^N {fleft( {z e^2 pi iomega left( n right)} right)} - fleft( 0 right)} right|, and hfill delta _N^r left( {omega ;f} right) = mathop {sup }limits_{left| z right| = r} left| {left( {{1 mathord{left/ {vphantom {1 N}} right. kern-nulldelimiterspace} N}} right)sumlimits_{n = 1}^N {fleft( {z omega left( n right)} right)} cdot z - intlimits_0^z {fleft( zeta right)dzeta } } right|, hfill end{gathered} $$ wheref is a holomorphic function defined in the unit disc withf ^(k) (0)≠0 for allk∈?,r<1 is a positive number, and ω is a sequence in [0, 1]. The first of these discrepancies can be generalized for multidimensional sequences. ω is uniform distributed if and only if lim _N→∞ Δ _N ^r (ω;f)=0 resp. lim _N→∞δ _N ^r (ω;f)=0. These results are proved in a quantitative way by estimating the classical discrepancyD _N (ω) by means ofΔ _N ^r (ω;f) and δ _N ^r (ω;f): $$begin{gathered} Delta _N^r left( {omega ;f} right) ll D_N left( omega right) ll Phi left( {Delta _N^r left( {omega ;f} right)} right), hfill delta _N^r left( {omega ;f} right) ll D_N left( omega right) ll Psi left( {delta _N^r left( {omega ;f} right)} right). hfill end{gathered} $$ The functions Φ and Ψ only depend onf andr. These estimations are based on the inequalities ofKoksma-Hlawka andErdös-Turán.