1. bookVolume 16 (2021): Edizione 1 (June 2021)
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Rivista
eISSN
2309-5377
Prima pubblicazione
30 Dec 2013
Frequenza di pubblicazione
2 volte all'anno
Lingue
Inglese
access type Accesso libero

Families of Well Approximable Measures

Pubblicato online: 30 Oct 2021
Volume & Edizione: Volume 16 (2021) - Edizione 1 (June 2021)
Pagine: 53 - 70
Ricevuto: 02 Sep 2020
Accettato: 13 Apr 2021
Dettagli della rivista
License
Formato
Rivista
eISSN
2309-5377
Prima pubblicazione
30 Dec 2013
Frequenza di pubblicazione
2 volte all'anno
Lingue
Inglese
Abstract

We provide an algorithm to approximate a finitely supported discrete measure μ by a measure νN corresponding to a set of N points so that the total variation between μ and νN has an upper bound. As a consequence if μ is a (finite or infinitely supported) discrete probability measure on [0, 1]d with a sufficient decay rate on the weights of each point, then μ can be approximated by νN with total variation, and hence star-discrepancy, bounded above by (log N)N1. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measure μ, there exist finite sets whose star-discrepancy with respect to μ is at most (logN)d12N1 {\left( {\log \,N} \right)^{d - {1 \over 2}}}{N^{ - 1}} . Moreover, we close a gap in the literature for discrepancy in the case d =1 showing both that Lebesgue is indeed the hardest measure to approximate by finite sets and also that all measures without discrete components have the same order of discrepancy as the Lebesgue measure.

Keywords

MSC 2010

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