BMO and exponential Orlicz space estimates of the discrepancy function in arbitrary dimension

In the current paper, we obtain discrepancy estimates in exponential Orlicz and BMO spaces in arbitrary dimension d ≥ 3. In particular, we use dyadic harmonic analysis to prove that the dyadic product BMO and exp(L2/(d−1)) norms of the discrepancy function of so-called digital nets of order two are bounded above by (logN)(d−1)/2. The latter bound has been recently conjectured in several papers and is consistent with the best known low-discrepancy constructions. Such estimates play an important role as an intermediate step between the well-understood Lp bounds and the notorious open problem of finding the precise L∞ asymptotics of the discrepancy function in higher dimensions, which is still elusive.

• Publication year

  2014

• Venue

  Journal d'Analyse Mathematique

• Publication date

  2014-11-21

• Fields of study

  Mathematics

• Identifiers
  DOI 10.1007/s11854-018-0035-xarXiv 1411.5794
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Discrepancy bounds for infinite-dimensional order two digital sequences over F2

  2014cited by this paper
• Dyadic shift randomization in classical discrepancy theory

  2014cited by this paper
• $L_p$- and $S_{p,q}^rB$-discrepancy of (order $2$) digital nets

  2014cited by this paper
• Dyadic shift randomization in classical discrepancy theory

  2014cited by this paper
• Discrepancy, Integration and Tractability

  2013cited by this paper
• Discrepancy and integration in function spaces with dominating mixed smoothness

  2013cited by this paper
• A two scale model for liquid phase epitaxy with elasticity: An iterative procedure

  2013cited by this paper
• On mean values of the _{}-discrepancies of point distributions

  2013cited by this paper
• Estimates of the Discrepancy Function in Exponential Orlicz Spaces

  2013cited by this paper
• Explicit constructions of point sets and sequences with low discrepancy

  2013cited by this paper
• Optimal $\mathcal{L}_2$ discrepancy bounds for higher order digital sequences over the finite field $\mathbb{F}_2$

  2012cited by this paper
• Quasi-Monte Carlo methods for integration of functions with dominating mixed smoothness in arbitrary dimension

  2012cited by this paper
• A Continuous Version of Duality of H 1 with BMO on the Bidisc

  2012cited by this paper
• ON ROTH’S ORTHOGONAL FUNCTION METHOD IN DISCREPANCY THEORY

  2011cited by this paper
• DISCREPANCY OF GENERALIZED HAMMERSLEY TYPE POINT SETS IN BESOV SPACES WITH DOMINATING MIXED SMOOTHNESS

  2011cited by this paper
• Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration

  2010influential reference
• On lower bounds for the L2-discrepancy

  2010cited by this paper
• L_2 discrepancy of generalized two-dimensional Hammersley point sets scrambled with arbitrary permutations

  2010cited by this paper
• Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration

  2010cited by this paper
• Discrepancy of Hammersley points in Besov spaces of dominating mixed smoothness

  2010cited by this paper
• Tractability of Multivariate Problems Volume II: Standard Information for Functionals

  2010cited by this paper
• Geometric Discrepancy: An Illustrated Guide

  2009cited by this paper
• Walsh Spaces Containing Smooth Functions and Quasi-Monte Carlo Rules of Arbitrary High Order

  2008cited by this paper
• BMO from dyadic BMO on the bidisc

  2008cited by this paper
• Orthogonality and Digit Shifts in the Classical Mean Squares Problem in Irregularities of Point Distribution

  2008cited by this paper
• Exponential Squared Integrability for the Discrepancy Function in Two Dimensions

  2008cited by this paper
• On the Small Ball Inequality in All Dimensions

  2007cited by this paper
• Explicit Constructions of Quasi-Monte Carlo Rules for the Numerical Integration of High-Dimensional Periodic Functions

  2007influential reference
• Harmonic analysis on totally disconnected groups and irregularities of point distributions

  2006cited by this paper
• Explicit constructions in the classical mean squares problem in irregularities of point distribution

  2002cited by this paper
• Monte Carlo and quasi-Monte Carlo methods

  1998cited by this paper
• IRREGULARITIES OF DISTRIBUTION (Cambridge Tracts in Mathematics 89)

  1988cited by this paper
• Irregularities of distribution

  1987cited by this paper
• Point sets and sequences with small discrepancy

  1987influential reference
• Some weighted norm inequalities concerning the schrödinger operators

  1985influential reference
• A Continuous Version of Duality of H 1 with BMO on the Bidisc

  1980cited by this paper
• On irregularities of distribution.

  1980cited by this paper
• On irregularities of distribution IV

  1979cited by this paper
• Uniform distribution of sequences

  1974cited by this paper
• Classical Banach spaces

  1973cited by this paper
• On irregularities of distribution vii

  1972cited by this paper
• Irregularities of distribution

  1968cited by this paper
• On the distribution of points in a cube and the approximate evaluation of integrals

  1967cited by this paper
• On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals

  1960cited by this paper
• Note on irregularities of distribution

  1956cited by this paper
• On irregularities of distribution

  1954influential reference

• The BMO-discrepancy suffers from the curse of dimensionality

  2023cites this paper
• Automated Measurement of Bone Thickness on SCT Sections and Other Images

  2020cites this paper
• Regularity results to semilinear Parabolic Initial Boundary Value Problems with Nonlinear Newton Boundary Conditions in a polygonal space-time cylinder

  2020cites this paper
• Nearest neighbor based conformal prediction

  2020cites this paper
• Tractability properties of the discrepancy in Orlicz norms

  2019cites this paper
• 4. Recent advances in higher order quasi-Monte Carlo methods

  2019cites this paper
• Discrepancy of Digital Sequences: New Results on a Classical QMC Topic

  2018cites this paper
• Optimal Lp-discrepancy bounds for second order digital sequences

  2017cites this paper
• A non-classical unital of order four with many translations

  2016cites this paper
• Discrepancy of second order digital sequences in function spaces with dominating mixed smoothness

  2016influential citation
• Optimal Lp-discrepancy bounds for second order digital sequences

  2016cites this paper