Abstract

We characterise the closure in C∞,(R, R) of the algebra generated by an arbitrary finite point-separating set of C∞functions. The description is local, involving Taylor series. More precisely, a function f ∈ C∞ belongs to the closure of the algebra generated by ψ1,...,ψr as soon as it has the 'right kind' of Taylor series at each point a such that ψ1'(a)...=ψr1 ( a)=0. The 'right kind' is of the form q 0 (T∞a ψ1 -ψ1(a), ..., T∞a ψr-ψr(a)), where q is a power series in r variables, and T a ψi denotes the Taylor series of ψi about a.