Integral representation of polynomial local functionals on convex functions

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.