An Archimedean Vector Lattice Functional Calculus For Semicontinuous Positively Homogeneous Functions

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.