Finitely-generated algebras of smooth functions, in one dimension (Q1269518)

Let \(\psi_1, \psi_2, \dots, \psi_n\) be \(n\) point-separating functions in \(C^\infty({\mathbb{R}})\), and let\linebreak \(R[\psi_1, \psi_2, \dots, \psi_n]\) denote the real algebra generated by them. The authors describe the closure of \(R[\psi_1, \psi_2, \dots, \psi_n]\) in \(C^\infty({\mathbb{R}})\). The main result says that if the mapping \(\langle\psi_1, \psi_2, \dots, \psi_n\rangle:{\mathbb{R}}\to{\mathbb{R}}^n\) is injective, then a function \(f\in C^\infty({\mathbb{R}})\) belongs to the closure of the algebra \(R[\psi_1, \psi_2, \dots, \psi_n]\) in \(C^\infty({\mathbb{R}})\) if and only if at any point \(a\) with \(\psi_1'(a)=\psi_2'(a)= \dots =\psi_n'(a)=0\) the Taylor series of \(f\) about \(a\) has the form \(q\circ(T_a^\infty \psi_1- \psi_1(a), \dots, T_a^\infty\psi_n-\psi_n(a))\), where \(q\) is a power series in \(n\) variables, and \(T_a^\infty\psi_i\) are the Taylor series of \(\psi_i\) about \(a\). This result answers the one-dimensional part of a general question asked by Segal for real subalgebras in \(C^\infty({\mathbb{R}}^d)\).