Finitely-Generated Algebras of Smooth Functions, in One Dimension

We characterise the closure inC∞(R, R) of the algebra generated by an arbitrary finite point-separating set ofC∞functions. The description is local, involving Taylor series. More precisely, a functionf∈C∞belongs to the closure of the algebra generated byψ1, …, ψras soon as it has the “right kind” of Taylor series at each pointasuch thatψ′1(a)=…=ψ′r(a)=0. The “right kind” is of the formq∘(T∞aψ1−ψ1(a), …, T∞aψr−ψr(a)), whereqis a power series inrvariables, andT∞aψidenotes the Taylor series ofψiabouta.