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The exact order of generalized diaphony and multidimensional numerical integration

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  09 April 2009

Vsevolod F. Lev
Vsevolod F. Lev
Affiliation:
Department of Mathematics, The University of Georgia, Athens GA 30605, USA e-mail: seva@math.uga.edu
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Abstract

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For a point set in the multidimensional unit torus we introduce an Lk-measure of uniformity of distribution, which for k=2 reduces to diaphony (and thus in this case essentially coincides with Weyl L2-discrepancy). For k ∈ [1, 2] we establish a sharp asymptotic for this new measure as the number of points of the set tends to infinity. Upper and lower-bound estimates are given also for k >2.

Keywords

Diaphonydiscrepancyuniform distribution

MSC classification

Secondary: 11K38: Irregularities of distribution, discrepancy 11K70: Harmonic analysis and almost periodicity
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 66 , Issue 1 , February 1999 , pp. 1 - 17
Copyright
Copyright © Australian Mathematical Society 1999

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