An extension of Warren's lower bounds for approximations (Q686108)

Let \(M\) be a compact topological space and \(K \subset C(M)\) a compact set. For \(F \subset C(M)\) one defines \(D(K,F)=\sup_{g \in K} \{\inf_{f \in F} \| f-g \|_ \infty\}\). Given \(\varphi:\mathbb{R}^ n \times M \to \mathbb{R}\) continuous, one associates \(F(\Phi) =\{\varphi(x,- ):M \to \mathbb{R}/x \in \mathbb{R}^ n\}\). One says that \(F(\Phi)\) has the Nash specialization of complexity \(\leq d\) if all members of \(F(\Phi)\) are Nash functions of complexity \(\leq d\). The main result of the paper gives a lower bound for \(\Delta_{n,d} (K)=\inf \{D(K,F(\Phi))\}\) where \(F(\Phi)\) runs over all families of functions with Nash specializations on \(R^ N\) of complexity \(\leq d\).