2021/10/08 by Dmitriy Bilyk, Bilyk, Dmitriy, Ryan Matzke +3
Mathematics · #49Q20 (Primary) 43A35 (Secondary) #52A40 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Approximation and Integration #math.CA #math.FA #msc:43A35 #msc:49Q20 #msc:52A40
paper · pdf · doi:10.48550/arxiv.2110.04138
30 pages, 1 figure
arxiv created 2021/10/08 · openalex publication_date 2021/10/08 · arxiv updated 2021/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we elaborate on the interplay between energy optimization, positive definiteness, and discrepancy. In particular, assuming the existence of a K-invariant measure μ with full support, we show that conditional positive definiteness of a kernel K is equivalent to a long list of other properties: including, among others, convexity of the energy functional, inequalities for mixed energies, and the fact that μ minimizes the energy integral in various senses. In addition, we prove a very general form of the Stolarsky Invariance Principle on compact spaces, which connects energy minimization and discrepancy and extends several previously known versions.