Approximation of Discrete Measures by Finite Point Sets

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 1 2N 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 this gap by giving a complete description of the discrete case. Most importantly, we prove that for any discrete measure the best possible order of approximation is for infinitely many N bounded from below by 1 cN for some constant 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 1 N is indeed the optimal one.