TY - JOUR
T1 - Positive Definiteness and the Stolarsky Invariance Principle
AU - Matzke, Ryan William
AU - Bilyk, Dmitriy
AU - Vlasiuk, Oleksandr
PY - 2022
Y1 - 2022
N2 - 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.
AB - 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.
U2 - 10.1016/j.jmaa.2022.126220
DO - 10.1016/j.jmaa.2022.126220
M3 - Article
SN - 0022-247X
VL - 513
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 2
M1 - 126220
ER -