Abstract
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.
| Original language | English (US) |
|---|---|
| Article number | 126220 |
| Journal | Journal of Mathematical Analysis and Applications |
| Volume | 513 |
| Issue number | 2 |
| DOIs | |
| State | Published - Sep 15 2022 |
Bibliographical note
Funding Information:The research of the first author (D. Bilyk) received partial funding from the National Science Foundation (grants DMS-1665007 and DMS-2054606 ) and the Simons Foundation (Collaboration Grant 712810 ). O. Vlasiuk has been supported by an AMS-Simons Travel grant for early career mathematicians. In addition, the authors were supported by ICERM ( Brown University ) and NSF for a research in groups meeting.
Publisher Copyright:
© 2022 Elsevier Inc.
Keywords
- Discrepancy
- Minimal energy
- Positive definiteness
- Stolarsky principle