<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>For a probability measure <italic>μ</italic> on [0, 1] without discrete component, the best possible order of approximation by a finite point set in terms of the star-discrepancy is &amp;inline as has been proven relatively recently. However, if <italic>μ</italic> contains a discrete component no non-trivial lower bound holds in general because it is straightforward to construct examples without any approximation error in this case. This might explain, why the approximation of discrete measures on [0, 1] by finite point sets has so far not been completely covered in the existing literature. In this note, we close the gap by giving a complete description for discrete measures. Most importantly, we prove that for any discrete measures (not supported on one point only) the best possible order of approximation is for infinitely many <italic>N</italic> bounded from below by &amp;inline for some constant 6 ≥ <italic>c></italic> 2 which depends on the measure. This implies, that for a finitely supported discrete measure on [0, 1]<sup>d</sup> the known possible order of approximation &amp;inline is indeed the optimal one.</p>
</abstract>

For a probability measure μ on [0, 1] without discrete component, the best possible order of approximation by a finite point set in terms of the star-discrepancy is &inline as has been proven relatively recently. However, if μ contains a discrete component no non-trivial lower bound holds in general because it is straightforward to construct examples without any approximation error in this case. This might explain, why the approximation of discrete measures on [0, 1] by finite point sets has so far not been completely covered in the existing literature. In this note, we close the gap by giving a complete description for discrete measures. Most importantly, we prove that for any discrete measures (not supported on one point only) the best possible order of approximation is for infinitely many N bounded from below by &inline for some constant 6 ≥ c> 2 which depends on the measure. This implies, that for a finitely supported discrete measure on [0, 1]d the known possible order of approximation &inline is indeed the optimal one.

Approximation of Discrete Measures by Finite Point Sets

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Approximation of Discrete Measures by Finite Point Sets"}

For a probability measure μ on [0, 1] without discrete component, the best possible order of approximation by a finite point set in terms of the...

Approximation of Discrete Measures by Finite Point Sets

Published Online: Aug 10, 2023

Page range: 31 - 38

Received: Jul 07, 2022

Accepted: Feb 09, 2023

DOI: https://doi.org/10.2478/udt-2023-0003

Keywords
Star-discrepancy, discrete measures, Borel measures

© 2023 Christian Weiss, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Approximation of Discrete Measures by Finite Point Sets

Published Online: Aug 10, 2023

Page range: 31 - 38

Received: Jul 07, 2022

Accepted: Feb 09, 2023

DOI: https://doi.org/10.2478/udt-2023-0003

KeywordsStar-discrepancy, discrete measures, Borel measures

© 2023 Christian Weiss, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Star-discrepancy, discrete measures, Borel measures