Discrepancy and Numerical Integration on Metric Measure Spaces

  • Luca Brandolini
  • William W. L. Chen
  • Leonardo Colzani
  • Giacomo Gigante
  • Giancarlo Travaglini
Article
  • 15 Downloads

Abstract

We study here the error of numerical integration on metric measure spaces adapted to a decomposition of the space into disjoint subsets. We consider both the error for a single given function, and the worst case error for all functions in a given class of potentials. The main tools are the classical Marcinkiewicz–Zygmund inequality and ad hoc definitions of function spaces on metric measure spaces. The same techniques are used to prove the existence of point distributions in metric measure spaces with small \(L^p\) discrepancy with respect to certain classes of subsets, for example, metric balls.

Keywords

Discrepancy Numerical integration Metric measure spaces 

Mathematics Subject Classification

65D30 11K38 

Notes

Acknowledgements

The authors wish to thank Dmitriy Bilyk for several conversations concerning the results on discrepancy contained in this paper.

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria Gestionale, dell’Informazione e della ProduzioneUniversità degli Studi di BergamoDalmineItaly
  2. 2.Department of MathematicsMacquarie UniversitySydneyAustralia
  3. 3.Dipartimento di Matematica e ApplicazioniUniversità di Milano–BicoccaMilanItaly

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