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Báo cáo toán học: "Bounds for the average Lp-extreme and the L∞-extreme discrepancy"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Bounds for the average Lp-extreme and the L∞-extreme discrepancy. | Bounds for the average Lp-extreme and the L1 -extreme discrepancy Michael Gnewuch Mathematisches Seminar Christian-Albrechts-Universitat Kiel Christian-Albrechts-Platz 4 D-24098 Kiel Germany e-mail mig@numerik.uni-kiel.de Submitted Jan 24 2005 Accepted Oct 18 2005 Published Oct 25 2005 Mathematics Subject Classifications 11K38 Abstract The extreme or unanchored discrepancy is the geometric discrepancy of point sets in the d-dimensional unit cube with respect to the set system of axis-parallel boxes. For 2 p 1 we provide upper bounds for the average Lp-extreme discrepancy. With these bounds we are able to derive upper bounds for the inverse of the L1-extreme discrepancy with optimal dependence on the dimension d and explicitly given constants. 1 Introduction Let Rd be the set of all half-open axis-parallel boxes in the d-dimensional unit ball with respect to the maximum norm i.e. Rd x y I x y 2 -1 l d X y where x y xi yi X. X xd yd and inequalities between vectors are meant componentwise. It is convenient to identify Rd with e x x 2 R2d I 1 X X 1 where for any real scalar a we put a a . a 2 Rd. The Lp-extreme discrepancy of a point set ti . tn c 1 1 d is given by d Dp ti . tn n X1 X1 1 it d x X 1 p V J 1 1 n i 1 Q 1 1 i i Supported by the Deutsche Forschungsgemeinschaft under Grant SR7 10-1 THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R54 1 where 1 denotes the characteristic function of x x and d is the normalized Lebesgue measure 2-ddxdx on Q. The L1 -extreme discrepancy is Di ti . tn sup Yx - Xz - 1 X 1 ti n and the smallest possible L1 -extreme discrepancy of any n-point set is D1 n d inf Di ti . tn . ti . t e -i i d Another quantity of interest is the inverse of D1 n d namely n1 e d min n G N I D1 n d . If we consider in the dehnitions above the set of all d-dimensional corners Cd 1 y 1 y G 1 1 d instead of Rd we get the classical notion of star-discrepancy. It is well known that the star-discrepancy is related to the error of multivariate integration of .