Potential theory with multivariate kernels

Damir Ferizović, Dmitriy Bilyk*, Ryan Matzke, Josiah Park, Alexey Glazyrin, Oleksandr Vlasiuk

*Corresponding author for this work

Research output: Working paper

Abstract

In the present paper we develop the theory of minimization for energies with multivariate kernels, i.e. energies, in which pairwise interactions are replaced by interactions between triples or, more generally, n-tuples of particles. Such objects, which arise naturally in various fields, present subtle differences and complications when compared to the classical two-input case. We introduce appropriate analogues of conditionally positive definite kernels, establish a series of relevant results in potential theory, explore rotationally invariant energies on the sphere, and present a variety of interesting examples, in particular, some optimization problems in probabilistic geometry which are related to multivariate versions of the Riesz energies.
Original languageEnglish
Number of pages23
Publication statusSubmitted - 9 Apr 2021

Keywords

  • Classical Analysis
  • Functional Analysis

ASJC Scopus subject areas

  • Analysis

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