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DISCREPANCY OF SECOND ORDER DIGITAL SEQUENCES IN FUNCTION SPACES WITH DOMINATING MIXED SMOOTHNESS

Part of: Probabilistic methods, simulation and stochastic differential equations Probabilistic theory: distribution modulo $1$; metric theory of algorithms Nontrigonometric harmonic analysis Holomorphic functions of several complex variables Linear function spaces and their duals

Published online by Cambridge University Press:  29 November 2017

Josef Dick ,
Aicke Hinrichs ,
Lev Markhasin  and
Friedrich Pillichshammer
Josef Dick
Affiliation:
School of Mathematics and Statistics, The University of New South Wales, Sydney NSW 2052, Australia email josef.dick@unsw.edu.au
Aicke Hinrichs
Affiliation:
Institut für Funktionalanalysis, Johannes Kepler Universität Linz, Altenbergerstraße 69, 4040 Linz, Austria email aicke.hinrichs@jku.at
Lev Markhasin
Affiliation:
Institut für Stochastik uand Anwendungen, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany email lev.markhasin@mathematik.uni-stuttgart.de
Friedrich Pillichshammer
Affiliation:
Institut für Finanzmathematik und angewandte Zahlentheorie, Johannes Kepler Universität Linz, Altenbergerstraße 69, 4040 Linz, Austria email friedrich.pillichshammer@jku.at
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Abstract

The discrepancy function measures the deviation of the empirical distribution of a point set in $[0,1]^{d}$ from the uniform distribution. In this paper, we study the classical discrepancy function with respect to the bounded mean oscillation and exponential Orlicz norms, as well as Sobolev, Besov and Triebel–Lizorkin norms with dominating mixed smoothness. We give sharp bounds for the discrepancy function under such norms with respect to infinite sequences.

MSC classification

Primary: 11K06: General theory of distribution modulo $1$ 11K38: Irregularities of distribution, discrepancy
Secondary: 32A37: Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 65C05: Monte Carlo methods
Type
Research Article
Information
Mathematika , Volume 63 , Issue 3 , 2017 , pp. 863 - 894
Copyright
Copyright © University College London 2017 

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