A universal approximation theorem and its applications to vector lattice theory (2507.20219v1)

Abstract: A classical result in approximation theory states that for any continuous function ( \varphi: \mathbb{R} \to \mathbb{R} ), the set ( \operatorname{span}{\varphi \circ g : g \in \operatorname{Aff}(\mathbb{R})} ) is dense in ( \mathcal{C}(\mathbb{R}) ) if and only if ( \varphi ) is not a polynomial. In this note, we present infinite dimensional variants of this result. These extensions apply to neural network architectures and improves the main density result obtained in \cite{BDG23}. We also discuss applications and related approximation results in vector lattices, improving and complementing results from \cite{AT:17, bhp,BT:24}.