An Archimedean Vector Lattice Functional Calculus For Semicontinuous Positively Homogeneous Functions

TL;DR

This paper introduces a new functional calculus for semicontinuous positively homogeneous functions on Archimedean vector lattices, extending previous continuous calculus and providing concrete formulas via saddle representations.

Contribution

It extends existing continuous functional calculus to semicontinuous functions and offers explicit formulas using saddle representations within vector lattices.

Findings

Extended functional calculus to semicontinuous functions

Provided concrete formulas via saddle representations

Illustrated utility with examples

Abstract

We develop a functional calculus on Archimedean vector lattices for semicontinuous positively homogeneous real-valued functions defined on $\R^n$ which are bounded on the unit sphere. It is further shown that this semicontinuous Archimedean vector lattice functional calculus extends the existing continuous Archimedean vector lattice functional calculus by Buskes, de Pagter, and van Rooij. We further utilize saddle representations of continuous positively homogeneous functions to provide concrete formulas, for functions abstractly defined via the continuous functional calculus, which are completely in terms of vector lattice operations. Finally, we provide some examples to illustrate the utility of the theory presented.