Integral representation of polynomial local functionals on convex functions

TL;DR

This paper develops integral representations for polynomial local functionals on convex functions, using approximation and distribution classification, with applications to Monge--Ampère-type operators.

Contribution

It introduces a novel integral representation framework for polynomial local functionals on convex functions, extending understanding of their structure and applications.

Findings

Established integral representations for polynomial local functionals.

Classified dense subspace of smooth polynomial local functionals.

Proved density results for Monge--Ampère-type operators.

Abstract

Integral representations for continuous polynomial local functionals on convex functions are established in terms of a finite family of polynomials. This result is obtained by approximation from a classification of the dense subspace of smooth polynomial local functionals, which is based on a Paley--Wiener--Schwartz-type classification of the Goodey--Weil distributions associated to these functionals under support restrictions. As an application, density results for various families of Monge--Amp\`ere-type operators are established.