Approximations in globally subanalytic and Denjoy-Carleman classes

Abstract

Efroymson's Approximation Theorem asserts that if f is a C0 semialgebraic mapping on a C∞ semialgebraic submanifold M of Rn and if ε:M→R is a positive continuous semialgebraic function then there is a C∞ semialgebraic function g:M→R such that |f−g|<ε. We prove a generalization of this result to the globally subanalytic category. Our theorem actually holds in a larger framework since it applies to every function which is definable in a polynomially bounded o-minimal structure (expanding the real field) that admits C∞ cell decomposition. We also establish approximation theorems for Lipschitz and C1 definable functions.