Approximation of Discrete Measures by Finite Point Sets

• Published in: Uniform distribution theory Vol. 18; no. 1; pp. 31 - 38
• Main Author: Weiss, Christian
• Format: Journal Article
• Language: English
• Published: Liverpool Sciendo 01.07.2023; De Gruyter Brill Sp. z o.o., Paradigm Publishing Services
• Subjects: 11J71; 11K31; 11K36; 11K38; Approximation; Borel measures; discrete measures; Star-discrepancy
• ISSN: 2309-5377; 1336-913X; 2309-5377
• DOI: 10.2478/udt-2023-0003

For a probability measure 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 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 bounded from below by &inline for some constant 6 ≥ 2 which depends on the measure. This implies, that for a finitely supported discrete measure on [0, 1] the known possible order of approximation &inline is indeed the optimal one.