Year: 2019
Type: preprint
Abstract: The irregularities of a distribution of $N$ points in the unit interval are often measured with various notions of discrepancy. The discrepancy function can be defined with respect to intervals of the form $[0,t)\subset [0,1)$ or arbitrary subintervals of the unit interval. In the former case, it is a well known fact in discrepancy theory that the $N$-element point set in the with the lowest $L_2$ or $L_{\infty}$ norm of the discrepancy function is the centered regular grid $$ Γ_N:=\left\{\frac{2n+1}{2N}: n=0,1,\dots,N-1\right\}. $$ We show a stronger result on the distribution of discrepancy functions of point sets in $[0,1]$, which basically says that the distribution of the discrepancy function of $Γ_N$ is in some sense minimal among all $N$-element point sets. As a consequence, we can extend the above result to rearrangement-invariant norms, including $L_p$, Orlicz and Lorentz norms. We study the same problem for the discrepancy notions with respect to arbitrary subintervals. In this case, we will observe that we have to deal with integrals of convolutions of functions. To this end, we prove a general upper bound on such expressions, which might be of independent interest as well.
Source: arXiv (Cornell University)
Authors: Kritzinger, Ralph, Passenbrunner, Markus
Institution: Johannes Kepler University of Linz
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Austrian Science Fund
Awards: G5749826759, G8199110932, G8550582057
OpenAlex ID: W2951329172