Bounded measurable simultaneous monotone approximation

Abstract

Let X = [a, b] be a closed bounded real interval. Let B be the closed linear space of all bounded real valued functions defined on X, and let M ⊆ B be the closed convex cone consisting of all monotone non-decreasing functions on X. For f, g ∈ B and a fixed positive w ∈ B, we define the so-called best L_∞-simultaneous approximant of f and g to be an element h* ∈ M satisfying

for all h ∈ M, where

We establish a duality result involving the value of d in terms of f, g and w only.

If in addition f, g and w are continuous, then some characterisation results are obtained.